How many six digit numbers can be formed using 3s and 5s only such that each number is a multiple of 6?
The Improving Mathematics Education in Schools (TIMES) Project
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Multiples, Factors and Powers
Number and Algebra : Module 19
Years : 7-8
PDF Version of module
Multiplication and division of whole numbers throw up many surprising things. This module encourages multiplicative thinking about numbers, and introduces ideas that are essential skills in fractions and algebra.
The ideas of this module are presented in purely arithmetical form, and no algebra is used except in some remarks that look forward to later work. The only numbers in the module are whole numbers, apart from the final paragraphs, where fractions are used so that the fifth index law can be presented in a more satisfactory form.
Students first meet the distinction between odd numbers and even numbers in early primary school, but it is useful everywhere in mathematics. Even numbers are multiples of 2, and more generally, multiples arise throughout mathematics and everyday life. The mass of a stack of bricks is a multiple of the mass of one brick. The number of pages in a packet of notebooks is a multiple of the number of pages in one notebook.
The factors of a number can be displayed using rectangular arrays. Some numbers, such as 30, can arise in many different ways as a product,
30 = 1 × 30 = 2 × 15 = 3 × 10 = 5 × 6 = 2 × 3 × 5,
whereas a number such as 31 can only be written trivially as the product 31 = 1 × 31. This idea leads to the classification of numbers greater than 1 as either prime or composite, and to a listing of all the factors of a number.
There are several groups of well-known divisibility tests that can check whether a number is a factor without actually performing the division. These tests greatly simplify the listing of factors of numbers.
Repeated addition leads to multiplication. Repeated multiplication in turn leads to powers, and manipulating powers in turn relies on five index laws. Powers are introduced in this module, together with four of the five index laws.
We are used to comparing numbers in terms of their size. The highest common factor (HCF) and lowest common multiple (LCM) allow us to compare numbers in terms of their factors and multiples. For example, when we look at 30 and 12, we see that they are both multiples of 6, and that 6 is the greatest factor common to both numbers. We also see that 60 is a multiple of both numbers, and that 60 is the lowest common multiple of them (apart from 0). The HCF and LCM are essential for fractions and later for algebra.
Odd and even numbers
Here is the usual definition of odd and even whole numbers.
Thus 10 is even and 11 is odd. We can demonstrate this by writing
10 = 5 + 5
11 = 5 + 5 + 1,
and we can illustrate this using arrays with two rows.
The array representing the even number 10 has the dots divided evenly into two equal rows of 5, but the array representing the odd number 11 has an extra odd dot left over.
When we write out the whole numbers in order,
the even and odd numbers alternate, starting with 0, which is an even number because
This pattern occurs in all sorts of common situations:
Indeed, our concept of the number 2 is so different from our conceptions of all other numbers that we even use different language. We divide a pie between two people, but among three people. We identify two alternatives, but three options. The word ‘doubt’ is related to the Latin ‘duo’, the word ‘two-faced’ means ‘liar’, and the traditional number of the devil is 2.
Students often come up with other possible definitions of even whole numbers:
What particular property of odd and even numbers does each illustrate?
Adding and subtracting odd and even numbers
There are several obvious facts about calculations with odd and even numbers that are very useful as an automatic check of calculations. First, addition and subtraction:
Proofs by arrays usually convince students more than algebraic proofs. The diagram below illustrates ‘odd plus odd equals even’, and shows how everything depends on the odd dot left over. The other cases are very similar.
Draw four diagrams to illustrate the four cases of subtraction of odd and even numbers.
Multiplication of odd and even numbers
When we multiply odd and even numbers,
Proofs by arrays can be used here, but they are unwieldy. Instead, we will use the previous results for adding odd and even numbers. Here are examples of the three cases:
6 × 4 = 24,
5 × 4 = 20,
7 × 3 = 21.
The first and second products are even because each can be written as the sum
6 × 4 = 4 + 4 + 4 + 4 + 4 + 4
5 × 4 = 4 + 4 + 4 + 4 + 4.
The third product can be written as the sum of pairs of odd numbers, plus an extra
7 × 3 = (3 + 3) + (3 + 3) + (3 + 3) + 3.
Each bracket is even, because it is the sum of two odd numbers, so the whole sum is odd.
What can we say about the quotients of odd and even numbers? Assuming in each case that the division has no remainder, complete each sentence below, if possible. Justify your answers by examples.a The quotient of an even number and an even number is…
bThe quotient of an even number and an odd number is…
c The quotient of an odd number and an even number is…
d The quotient of an odd number and an odd number is…
Using algebra for odd and even numbers
The previous results on the arithmetic of odd and even numbers can be obtained later after pronumerals have been introduced, and expansions of brackets and taking out a common factor dealt with. The important first step is:
Obtain the previous results on the addition and multiplication of odd and even numbers by algebra. Begin, ‘Let the even numbers be 2a and 2b, and the odd numbers be 2a + 1 and 2b + 1, where a and b are whole numbers.’
Representing numbers by arrays
In the previous section, we represented even numbers by arrays with two equal rows, and odd numbers by arrays with two rows in which one row has one more dot than the other.
Representing numbers by arrays is an excellent way to illustrate some of their properties. For example, the arrays below illustrate significant properties of the numbers 10, 9, 8 and 7.
The number 10
There are two rectangular arrays for 10:
The first array shows that 10 can be factored as 10 = 5 × 2, which means that 10 is an even number.
The second array is trivial − every number can be factored as a product of itself and 1.
The convention used in these modules is that the first factor represents the number of rows, and the second factor represents the number of columns. The opposite convention, however, would be equally acceptable.
The number 9
The number 9 also has two rectangular arrays:
The first array shows that 9 is a square because it can be represented as a square array. The corresponding factoring is 9 = 3 × 3.
Because there is no 2-row array, the number 9 is odd. The second array is the trivial array.
The number 8
The number 8 has two rectangular arrays and a three-dimensional array:
The third array shows that 8 is a cube because it can be represented as a cubic array.
The first array shows that 8 is even, and the second array is the trivial array.
The number 7
The interesting thing about the number 7 is that it only has the trivial array, because 7 dots cannot be arranged in any rectangular array apart from a trivial array.
Numbers greater than 1 with only a trivial rectangular array are called prime numbers. All other whole numbers greater than 1 are called composite numbers.
We shall discuss prime numbers in a great deal more detail in the later module, Primes and Prime Factorisation. The usual definition of a prime number expresses exactly the same thing in terms of factors:
Here are the only possible rectangular arrays for the first four prime numbers:
Exercise 5aDraw all possible rectangular arrays for the numbers from 1 to 12, including any three dimensional arrays, and identify the corresponding factorings. Which of these numbers are prime, which are composite, which are square, and which are even?
bList all the prime numbers less than 50. How many of these are even?
cIdentify all two-dimensional and three-dimensional arrays for 36.
dDraw all rectangular arrays for 16, including three-dimensional arrays.
eIf you had four dimensions at your disposal, what array would you draw?
fWhat is the smallest number greater than 1 that is both square and cubic?
What are the side lengths of the corresponding arrays?
gWhy is a manufacturer more likely to pack his goods in packets of 30 than in packets of 29 or 31?
EXERCISE 6a Prove that when the diagonal of a square array is removed, the number of remaining dots is even.
bIf a number is subtracted from its square, what sort of number remains?
EXERCISE 7a Prove that in any square array, the number of dots on the outside of the array
is a multiple of 4.
b Hence prove that the difference of two even squares is divisible by 4, and that the difference of two odd squares is divisible by 4.
Rectangular arrays are not the only way that numbers can usefully be represented by patterns of dots.
Thus the first few triangular numbers are: 1, 3, 6, 10, 15, 21,…
EXERCISE 8aExplain the pattern of differences between successive triangular numbers.
bHence write down the first 20 triangular numbers.
cIdentify and explain the pattern of odd and even numbers in this sequence.
Explain how to calculate from this diagram that the 4th triangular number is 10. Hence calculate the 100th triangular number.
Multiples, common multiples and the LCM
We can arrange the multiples of 6 in increasing order,
0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96,…
so that they form a simple pattern increasing by 6 at each step.
Because 6 is an even number, all its multiples are even. The multiples of an odd number such as 7, however, alternate even, odd, even, odd, even,… because we are adding the odd number 7 at each step.
The number zero is a multiple of every number. Because of this, the word ‘multiple’ is sometimes used in the sense of ‘nonzero multiple’, but we will always add the word ‘nonzero’ when 0 is excluded.
The multiples of zero are all zero. Every other whole number has infinitely many multiples. This phrase ‘infinitely many’ has a very precise meaning — however many multiples you write down, there is always another multiple that you haven’t written down.
We can illustrate the multiples of a number using arrays with three columns and an increasing numbers of rows. Here are the first few multiples of 3:
Rows and columns can be exchanged. Thus the multiples of 3 could also be illustrated using arrays with three rows and an increasing numbers of columns.
The repeating pattern of common multiples is a great help in understanding division.
0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96,…
If we divide any of these multiples by 6, we get a quotient with remainder zero.
24 ÷ 6 = 4
30 ÷ 6 = 5.
To divide any other number such as 29 by 6, we first locate 29 between two multiples of 6. Thus we locate 29 between 24 and 30. Because 29 = 24 + 5, we write
29 ÷ 6 = 4 remainder 5
29 = 6 × 4 + 5.
Notice that the remainder is always a whole number less than 6, because the multiples of 6 step up by 6 each time. Hence with division by 6 there are only 6 possible remainders,
0, 1, 2, 3, 4, 5.
This result takes a very simple form when we divide by 2, because the only possible remainders are 0 and 1.
In later years, when students have become far more confident with algebra, these remarks about division can be written down very precisely in what is called the division algorithm.
For example, we saw that 29 ÷ 6 = 4 remainder 5 means that
29 = 6 × 4 + 5
0 £ 5 < 6.
The table above shows all the whole numbers written out systematically in 7 columns. Suppose that each number in the table is divided by 7 to produced a quotient and
bWhat is the same about the results of the division in each column?
When 22 and 41 are divided by 6, their remainders are 4 and 5 respectively. Yet when their sum 63 is divided by 6, the remainder is 3, and is not 4 + 5 = 9. Explain.
Common multiples and the LCM
An important way to compare two numbers is to compare their lists of multiples. Let us write out the first few multiples of 4, and the first few multiples of 6, and compare the two lists.
The numbers that occur on both lists have been circled, and are called common multiples.
The common multiples of 6 and 8 are 0, 12, 24, 36, 48,…
Apart from zero, which is a common multiple of any two numbers, the lowest common multiple of 4 and 6 is 12.
These same procedures can be done with any set of two or more non-zero whole numbers.
EXAMPLEa Write out the first few common multiples of 12 and of 16. Hence write out the first few common multiples of 12 and 16, and state their lowest common multiple.
b Write out the first few multiples of 24. Hence write down the LCM of 12, 16 and 24?
c What is the LCM of 12 and 15?
d What is the LCM of 4, 6 and 9?
Solutiona The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144,…
The multiples of 16 are 16, 32, 48, 64, 80, 96, 112, 128, 144, 160,…
Hence the common multiples of 12 and 16 are 48, 96, 144,… and their LCM is 48.
b The multiples of 24 are 0, 24, 48,… Hence the LCM of 12, 16 and 24 is 48.
c The LCM of 12 and 15 is 60.
d The LCM of 4, 6 and 9 is 36.
Two or more nonzero numbers always have a common multiple — just multiply the numbers together. But the product of the numbers is not necessarily their lowest common multiple. For example, in the case above of 4 and 6, one common multiple is 4 × 6 = 24, but their lowest common multiple is 12.
EXAMPLEa What is the LCM of 9 and 10?
bWhat is the LCM of the first five primes?
cWhat is the LCM of 6 and 24? What is the general situation illustrated here?
dWhat is the LCM of 1 and 14? What is the general situation illustrated here?
Solutiona The LCM of 9 and 10 is their product 90.
bThe LCM of 2, 3, 5, 7 and 11 is their product, which is 2310.
cBecause 24 is a multiple of 6, the LCM of 6 and 24 is 24.
dEvery number is a multiple of 1, so the LCM of 1 and 14 is 14.
The common multiples are the multiples of the LCM
You will have noticed that the list of common multiples of 4 and 6 is actually a list of multiples of their LCM 12. Similarly, the list of common multiples of 12 and 16 is a list of the multiples of their LCM 48.
This is a general result, which in Year 7 is best demonstrated by examples. In an exercise at the end of the module, Primes and Prime Factorisation, however, we have indicated how to prove the result using prime factorisation.
We will have more to say about the LCM once the HCF has been introduced.
Factors, common factors and the HCF
This can be restated in terms of the multiples of the previous section:
The number zero is a multiple of every number, so every number is a factor of zero. On the other hand, zero is the only multiple of zero, so zero is a factor of no numbers except zero. These rather odd remarks are better left unsaid, unless students insist. They should certainly not become a distraction from the nonzero whole numbers that we want to discuss.
The product of two nonzero whole numbers is always greater than or equal to each factor in the product. Hence the factors of a nonzero number like 12 are all less than or equal to 12. Thus whereas a positive whole number has infinitely many multiples, it has only finitely many factors.
The long way to find all the factors of 12 is to test systematically all the whole numbers less than 12 to see whether or not they go into 12 without remainder. The list of factors of 12 is
1, 2, 3, 4, 6 and 12.
It is very easy to overlook factors by this method, however. A far more efficient way, is to look for pairs of factors whose product is 12. Begin by testing all the whole numbers 1, 2, … that could be the smaller of a pair of factors with product 12,
1, 2, 3,…
We stop at 3 because 42 = 16 is greater than 12, so 4 cannot be the smaller of a pair.
We can display these pairs of factors by writing the 12 dots in all possible rectangular arrays:
For a larger number such as 60, the method recommended here has the following steps:
Thus the factors of 60 are
Use this method to write down all the factors of 72 and 160.
Common factors and the HCF
Another important way to compare two numbers is to compare their lists of factors.
The numbers that occur on both lists have been circled, and are called the common factors.
The common factors of 18 and 30 are 1, 2, 3 and 6.
The highest common factor is 6.
As with common multiples, these procedures can be done with any list of two or more whole numbers.
The HCF is also known as the ‘greatest common divisor’, with corresponding initials GCD.
EXAMPLEa Write out the factors of 30 and of 75. Hence write out the common factors
of 30 and 75, and their highest common factor.
b Write out the factors of 45. Hence write down the HCF of 30, 75 and 45.
cWhat is the HCF of 60, 80, 90 and 100?
dWhat is the HCF of 82 and 102?
SolutionaThe factors of 30 are 1, 30, 2, 15, 3, 10, 5, 6.
The factors of 75 are 1, 75, 3, 25, 5, 15.
Hence the common factors of 30 and 75 are 1, 3, 5, 15, and their HCF is 15.
bThe factors of 45 are 1, 3, 5, 9, 15, 45.
cHence the HCF of 30, 75 and 45 is 15.
dThe HCF of 60, 80, 90 and 100 is 10.
eThe HCF of 82 and 102 is 22.
Any collection of whole numbers always has 1 as a common factor. The question is whether the numbers have common factors greater than 1.
Every whole number is a factor of 0, so the common factors of 0 and say 12 are just the factors of 12, and the HCF of 0 and 12 is 12. A nonzero whole number has only a finite number of factors, so it has a greatest factor. Two or more numbers always have a HCF because at least one of them is nonzero. These are distractions from the main ideas.
EXAMPLEa What is the HCF of 6 and 24? What is the general situation illustrated here?
bWhat is the HCF of 1 and 14? What is the general situation illustrated here?
cWhat is the HCF of 21 and 10? Find three other pairs of composite numbers with the same HCF.
dWhat is the HCF of 0 and 15?
Solutiona Because 6 is a factor of 24, the HCF of 6 and 24 is 6.
bThe only factor of 1 is 1, so the HCF of 1 and 14 is 1.
cThe HCF of 21 and 10 is 1. Some other pairs of composite numbers with HCF 1 are 4 & 9, 16 & 25, 22 & 35, 14 & 15, 12 & 55.
dAll the factors of 15 are also factors of 0, so the HCF of 0 and 15 is 15.
The common factors are the factors of the HCF
You will have noticed that the list of common factors of 18 and 30 is actually a list of factors of their HCF 6. Similarly, the list of common factors of 30 and 75 is a list of the factors of their HCF 15.
Again, this is a general result, which in Year 7 is best demonstrated by examples. An exercise at the end of the module Primes and Prime Factorisation indicates how to prove the result using prime factorisation.
Two relationships between the HCF and LCM
The two relationships below between the HCF and the LCM are again best illustrated by examples in Year 7, but an exercise in the module Primes and Prime Factorisation indicates how they can be proven.
The first relationship is extremely useful, and is used routinely when working with common denominators of fractions.
The converse is also true, although it does not arise so often.
The numbers 4 and 9 have HCF 1, and their LCM is their product 36.
The numbers 6 and 7 have HCF 1 and their LCM is their product 42.
EXAMPLEa Find the HCF and LCM of 21 and 10.
bFind the HCF and LCM of 14 and 15.
SolutionaThe HCF of 21 and 10 is 1, and their LCM is their product 210.
b The HCF of 28 and 15 is 1, and their LCM is their product 420.
The second relationship is not so obvious, and needs to be brought out by examples.
For example, the numbers 4 and 6 have HCF 2 and LCM 12, and 2 × 12 = 4 × 6.
Confirm this relationship for:
12 and 9
40 and 10
Solutiona The numbers 10 and 15 have HCF 5 and LCM 30, and 10 × 15 = 5 × 30.
bThe numbers 12 and 9 have HCF 3 and LCM 36, and 12 × 9 = 3 × 36.
cThe numbers 40 and 10 have HCF 10 and LCM 40, and 10 × 40 = 40 × 10.
The three-part example below indicates how this relationship can be proven from the relationship above, although such a proof would be unsuited for most Year 7 students.
EXERCISE 13aFind the HCF and LCM of 12 and 20, and show that LCM × HCF = 12 × 20.
bShow that when 12 and 20 are each divided by their HCF, the HCF of the resulting numbers is 1.
c Show that when the LCM is divided by 12 and by 20, the HCF of the resulting
numbers is 1.
When we multiply a number by itself, we usually use a more concise notation,
The terms ‘3 squared’ for 32 and ‘3 cubed’ for 33 come from geometry. As we saw earlier in the module, we can arrange 32 dots in a square and 33 dots in a cube:
before, there is no straightforward geometrical representation of
Exercise 14a Write down the first 16 squares, starting from 02 = 0.
b Write down the first 13 cubes, starting from 03 = 0.
Exercise 15aWhat is the smallest number that can be written as the sum of two squares in two distinct ways?
bWhat is the smallest number that can be written as the sum of two nonzero squares
in two distinct ways?
cWhat is the smallest number that can be written as the sum of two distinct nonzero squares in two distinct ways?
dThe number 1729 is the smallest number that can be written as the sum of two cubes in two distinct ways. Show how this can be done. (This observation has become famous because of a conversation between the Indian mathematician Ramanujan and the English mathematician Hardy. See Wikipedia for the details.)
Adding successive odd numbers gives successive squares:
Dissect a square array of 25 dots to show why this is happening.
[A rather difficult challenge activity] Adding successive cubes gives the squares of the triangular numbers:
b Place the stack of 13 = 1 blocks on the table, then dismantle the 23 = 8 stack and reassemble it systematically around the 13 = 1 stack to produce a square 3 × 3 array.
c Dismantle the 33 = 27 stack and reassemble it systematically around the 3 × 3 array to produce a square 6 × 6 array.
d Explain how the pattern continues to work.
Successive powers of a number
Powers of numbers are used extensively later in the study of logarithms and of combinatorics. It is useful to be able to compute or remember some smaller powers quickly, and recognise them.
The powers of 2 are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192,…
The powers of 3 are: 3, 9, 27, 81, 243, 729,…
The powers of 4 are every second power of 2.
The powers of 5 are: 5, 25, 125, 625, 3125,…
The powers of 6 are: 6, 36, 216,…
The powers of 7 are: 7, 49, 343,…
The powers of 8 are every third power of 2.
The powers of 9 are every second power of 3.
The powers of 10 are: 10, 100, 1000, 10 000, 100 000, 1 000 000,…
The powers of 16 are every fourth power of 2.
Our base 10 place-value system displays every number as a sum of multiples of powers of 10.
Computers use a base 2 place-value system, so computer programmers need to know the powers of 2, and must be able to convert numbers quickly to a sum of powers of 2.
Write 201 as a sum of powers of 2. Hence write 201 in base 2 notation.
Test for divisibility
There are several straightforward tests for divisibility that are very useful when factoring numbers. They all have their origin in the base 10 that we use for our system of numerals.
Divisibility by powers of 2 and powers of 5
The tests for divisibility by powers of 2 and 5 arise from the fact that 10 = 2 × 5.
Because 10 is a multiple of 2, every multiple of 10 is a multiple of 2. Thus to test whether a number is divisible by 2, we only need to look at the last digit.
Similarly, 10 is a multiple of 5, so every multiple of 10 is a multiple of 5. Thus to test whether a number is divisible by 5, we only need to look at the last digit.
The number 864 ends with 4, so it is divisible by 2, but not by 5.
The number 1395 ends with 5, so it is divisible by 5, but not by 2.
The number 74 830 ends with 0, so it is divisible both by 2 and by 5.
The squares 22 = 4 and 52 = 25 are factors of 100, so every multiple of 100 is a multiple of 4 and of 25. Thus to test for divisibility by 4 and 25, we only need to look at the last two digits.
Similarly, 23 = 8 and 53 = 125 are factors of 1000, so
These tests can be continued for higher powers of 2 and 5.
Examplea Test 6 945 732 for divisibility by 2, 4, 8,…
b Test 5 671 625 for divisibility by 5, 25, 125,…
SolutionaThe number 6 945 732 is divisible by 4 (and hence also by 2) because 32 is divisible by 4, but it is not divisible by 8 because 732 is not divisible by 8.
b The number 5 671 625 is divisible by 125 (and hence also by 25 and 5) because 625 is divisible by 125, but it is not divisible by 54 = 625 because 1625 is not divisible by 625.
Divisibility by 9 and 3
The tests for divisibility by 9, and in turn by 3, arise from the fact that 9 is 1 less than 10.
When 1 or any power of 10 is divided by 9, the remainder is 1. For example,
Multiplying each by a single digit number less than 9,
The remainder when 9254 is divided by 9 is the same as the remainder when
Because 9 is a multiple of 3, the remainders after division by 3 follows a similar pattern,
4 and 4 have the same remainder after division by 3.
50 and 5 have the same remainder after division by 3.
200 and 2 have the same remainder after division by 3.
9000 and 9 have the same remainder after division by 3.
Adding these results, 9254 and 9 + 2 + 5 + 4 = 20 have the same remainder when divided by 3. The general result is
Find the remainders when 62 574 is divided by 9 and by 3.
The sum of the digits is 6 + 2 + 5 + 7 + 4 = 24, and 2 + 4 = 6.
With most students in Years 7−8, however, only the corresponding divisibility tests
Test for divisibility by 3 and by 9:
Solutiona The sum of the digits is 7 + 1 + 3 + 2 + 5 + 6 + 1 + 8 = 33, and 3 + 3 = 6.
Hence 71 325 618 is divisible by 3, but not by 9.
bThe sum of the digits is 3 + 6 + 8 + 6 + 7 + 9 + 2 + 4 = 45, and 4 + 5 = 9.
Hence 36 867 924 is divisible by 3 and by 9.
Divisibility by 11
The test for divisibility by 11 arises from the fact that 11 is 1 more than 10, but 9 × 11 = 99 is 1 less than 100.
When 1 or any power of 10 is divided by 11, the remainder is alternately 1 and 10, but we shall regard the remainder 10 as being remainder −1.
Multiplying each result by a single digit number,
Adding these results, the remainder when 893 524 is divided by 11 is the same as the remainder when 4 − 2 + 5 − 3 + 9 − 8 = 7 is divided by 9. It was important here to write the alternating sum of the digits starting at the right-hand end with the units. The general result is,
91 627 ÷ 11 = 8329 remainder 8 and (7 − 2 + 6 − 1 + 9) ÷ 11 = 1 remainder 8.
Find the remainders after division by 11 of:
Solutiona The alternating sum of the digits is 8 − 7 + 5 − 2 + 6 = 10. Hence 62 578 has remainder 10 when divided by 11.
bThe alternating sum of the digits is 5 − 6 + 5 − 7 + 3 − 9 = −9. Hence 937 565 has remainder 11 − 9 = 2 when divided by 11.
As with 3 and 9, only the corresponding divisibility test is usually appropriate in Years 7−8:
Test for divisibility by 11:
Solutiona The alternating sum of the digits is 8 − 1 + 6 − 5 + 2 − 3 + 1 − 7 = 1.
Hence 71 325 618 is not divisible by 11.
bThe alternating sum of the digits is 1 − 7 + 4 − 8 + 3 − 9 + 1 − 7 = −22.
Hence 71 938 471 is divisible by 11.
Divisibility by numbers that are not prime powers
We can combine the previous tests for divisibility by testing separately for divisibility by the highest power of each prime. Here are examples of some such tests:
Divisibility by powers of 10 is particularly simple—just count the number of zeroes.
Examplea Test 726 for divisibility by 6, 12 and 18.
bTest 6875 for divisibility by 55 and 275.
Solutiona Factoring the three divisors, 6 = 2 × 3, 12 = 4 × 3 and 18 = 2 × 9. By looking at the last digit and the last two digits, 726 is divisible by 2, but not by 4.
The sum of the digits is 15, so 726 is divisible by 3, but not by 9.
bFactoring the two divisors, 55 = 5 × 11 and 275 = 25 × 11. By looking at the last digit and the last two digits, 6875 is divisible by 5 and by 25.
cThe alternating sum of the digits is 5 − 7 + 8 − 6 = 0, so 6875 is divisible by 11.
Hence 6875 is divisible by 55 and by 275.
The multiplication table
The multiplication table is one of the best-known objects in arithmetic. It is formed by doing nothing more that writing out the first twelve non-zero multiples of each number from 1 to 12 in twelve successive rows. (Or perhaps they were written out in twelve successive columns.)
Despite the simplicity of its construction, it is a very powerful object indeed, and well justifies the recommendation to learn it by heart. Here are some of its many properties — students routinely find many more.
Students often see many more patterns in this table. The following exercise gives some less obvious properties, but the proofs are omitted, because they require quite serious algebra. Once sequences and series are being studied, perhaps in Year 11, the table is well worth revisiting because of the insights it can give into sequences and into the use of algebra.
Exercise 19aWhat pattern is formed by shading the numbers that are multiples of 4?
bWhat pattern is formed by the differences between successive terms
on any diagonal parallel to the leading diagonal?
cNow take the opposite diagonal passing through the square 49 from top right
to bottom left. What numbers arise when we subtract each from 49?
What happens with parallel diagonals through other squares?
The index laws
There are five useful laws, collectively called the index laws, that help us manipulate powers. At this stage, they are best established by examples and learnt verbally.
After algebra has been introduced, however, they can be presented in their more
Multiplying powers with the same base
If we write out the powers as continued products, we can quickly see what is happening when we multiply powers with the same base.
75 × 73 = (7 × 7 × 7 × 7 × 7) × (7 × 7 × 7) = 78
There are 5 + 3 = 8 factors of 7 in the product, so the product is 78.
Dividing powers with the same base
We can use the same approach with dividing powers with the same base.
75 ÷ 73 = (7 × 7 × 7 × 7 × ) ÷ (7 × 7 × 7) = 72
There are 5 − 3 = 2 factors of 7 in the quotient, so the quotient is 72.
The examples above have carefully avoided questions like 73 ÷ 75, where the divisor is a higher power than the dividend. Such questions are better written as fractions,
The first form can be covered once fractions have been introduced. The second form requires negative indices, which are usually regarded as a little too sophisticated for Years 7−8.
Raising a power to a power
We often need to deal with expressions such as (73)4 in which a power is raised to a power. Writing out just the outer power allows us to apply the previous law.
There are 3 × 4 factors of 7 in the expression, so it simplifies to 712.
(73)4 = 712
Simplify each expression, leaving the answer as a power:
(42)3 × (43)2
(115)4 ÷ (116)3
The power of a product
The fourth law deals with a power of a product. Again, we can write out the power.
These steps rely heavily on the any-order property of multiplication to regroup the
Write each expression without brackets, leaving the answer in index notation.
a (3 × 7)4
d (5 × 7 × 9)6 ÷ (55)2
The power of a quotient
The final index law deals with the power of a quotient. It is clumsy to formulate this law using the division sign, so fraction notation for division is used for the only time in this module.
The same process can be done with the power of any fraction, giving the general result,
The index laws and mental arithmetic
‘A power of a product is the product of powers’ is useful in mental arithmetic, both forwards and in reverse. For example,
Because our system of numerals has base 10, isolating powers of 10 is always good practice. In the first example, we factored 20 as 20 = 2 × 10. In the second example, we combined factors of 2 and factors of 5 to make factors of 10.
The HCF and LCM are essential for the study of fractions, which are introduced in the module Fractions. To express a fraction in simplest form, we cancel the HCF of the numerator and denominator, and to add and subtract fractions, we usually use the LCM of their denominators as a common denominator,
The HCF of two algebraic expressions is important in factoring. The first step in the systematic factoring of an expression is to take out the HCF of the terms:
3ax2 − 6a2x = 3ax(x − 2a).
Factoring by taking out the HCF is first considered in the module Algebraic Expressions, and developed further in the module, Negatives and the Index Laws.
Prime numbers are the building blocks of all whole numbers when we take them apart by factoring. For example, we can write 60 as a product of primes,
60 = 22 × 3 × 5.
Every composite number has one and only one prime factorisation. Prime factorisation is the central concern of the module Primes and Prime Factorisation. This module also shows how prime factorisation yields more systematic approaches to the calculations of the HCF and LCM.
The five index laws have been introduced in the present module in the context of whole numbers, although it was mentioned that the fifth law, and a clearer statement of the second law, require fractions. The modules, Negatives and the Index Laws and Special Expansions and Algebraic Fractions extend them to rational numbers and also give them an algebraic formulation, although the indices are still only nonzero whole numbers. The later module The Index Laws extends the indices to any rational number.
The presentation in this module has been carried out arithmetically, and entirely within the whole numbers. There have been no algebra or fractions except in the occasional remark and exercise that looks forward to later content. Arrays, rather than areas, have been used to represent products, so that the whole-number aspect of the ideas can be emphasised. It is important in later years, when algebra has been used to generalise the content, not to lose the arithmetic intuition of the present module.
All the present material was expounded by the ancient Greek mathematicians. The idea of representing numbers by arrays was developed particularly by Pythagoras and his school, who developed theories about what we now call ‘figurate numbers’. These include the triangular numbers discussed in this module, and pentagonal and tetrahedral numbers.
ANSWERS TO EXERCISES
Here is ‘odd minus odd equals even’. The other cases are similar.
Exercise 3aThe quotient of an even number and an even number may be even or odd.
For example, 12 ÷ 6 = 2 and 12 ÷ 4 = 3.
bThe quotient of an even number and an odd number is always even.
For example, 12 ÷ 3 = 4.
c(When an odd number is divided by an even number, the remainder is never zero.)
d The quotient of an odd number and an odd number is always odd. For example,
35 ÷ 5 = 7.
Exercise 5aAll the numbers have a trivial array.
The number 1 and the primes 2, 3, 5, 7 and 11 each have only a trivial array.
The composite numbers 4, 6, 8, 9 and 12 each have at least one other array.
The even numbers 2, 4, 6, 8, 10 and 12 are those with a 2-row array.
The squares 4 and 9 also have a square array.
The numbers 6 and 10 each have only one non-trivial rectangular array.
The number 8 has the three-dimensional array and a 2 × 4 array.
The number 12 has three two-dimensional arrays because 12 = 1 × 12 = 2 × 6 = 3 × 4, and also has a three-dimensional 2 × 2 × 3 array.
bThe 15 primes less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
The number 2 is the only even prime number.
cThere are five two-dimensional arrays 36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6.
There are three three-dimensional arrays 36 = 2 × 2 × 9 = 2 × 3 × 6 = 3 × 3 × 4.
dThere are four arrays 16 = 1 × 16 = 2 × 8 = 4 × 4 = 2 × 2 × 4. If we had four dimensions at our disposal, we could draw the array corresponding to 16 = 2 × 2 × 2 × 2.
fThe smallest number greater than 1 that is both square and cubic is 64 = 8 × 8 = 4 × 4 × 4.
If we had use of six dimensions, we could draw the array 64 = 2 × 2 × 2 × 2 × 2 × 2.
gThere are a great many ways to arrange 30 objects in a rectangular pattern or three dimensional block with no wasted space. The numbers 29 and 31 are prime, so except for packing all the items in a row, any packing into a block will have empty spaces.
Exercise 6aThe dots in the lower left triangle of the square array are the reflection of the dots in the upper right triangle.
bIt follows immediately from part a that the difference is even. Alternatively, the square of an even number is even and the square of an odd number is odd, and the difference of two even numbers or two odd numbers is even.
Exercise 7aIf each side of the square has, for example 7 dots, then the total numbers of dots is
6 + 6 + 6 + 6 = 4 × 6.
bTake the larger of the two arrays, and keep removing its outer layer until it becomes the smaller array. At each step, the side length of the array goes down by 2, and the number of dots removed is a multiple of 4.
Exercise 8aThe difference increases by 1 each time, because each new row has one more dot.
b1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210,…
c Two odds, then two evens, then two odds, then two evens,…
Every time an odd number is added, the sum changes from even to odd or from odd to even. When an even number is added, however, the sum either stays even or stays odd.
The diagram shows that the 4th triangular number is (4 × 5) ÷ 2 = 10.
Similarly, the 100th triangular number is (100 × 101) ÷ 2 = 5050.
Exercise 10aAll the numbers except for the last number in the row have the same quotient. For example, the entries in the 5th row have quotient 4 when divided by 7.
bAll the numbers in, for example, the 5th column have remainder 5 when divided by 7.
The remainders when 22 and 41 are divided by 6 are 4 and 5, whose sum is 9.
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
The factors of 160 are 1, 2, 4, 5, 8, 10, 16, 20, 32 40, 80, 160.
Exercise 13aThe numbers 12 and 20 have HCF 4 and LCM 60, and 4 × 60 = 12 × 20.
bWhen 12 and 20 are divided by their HCF, which is 4, the resulting numbers are
3 and 5, and the numbers 3 and 5 have HCF 1.
cWhen the LCM, which is 60, is divided by 12 and by 20, the resulting numbers are
5 and 3, and the numbers 5 and 3 have HCF 1.
(Notice that the resulting numbers in parts b and c are the same.)
Exercise 14a0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225,…
b0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728,…
Exercise 15a25 = 52 + 02 = 32 + 42.
b50 = 52 + 52 = 72 + 12.
c65 = 82 + 12 = 72 + 42.
d1729 = 123 + 13 = 103 + 93.
201 = 128 + 64 + 8 + 1 = 27 + 26 + 23 + 1, so in base 2 notation, 201 = 11 001 001.
Exercise 19aEvery fourth row and every fourth column is shaded, together with the numbers in the exact centre of the remaining squares.
bThe differences along each diagonal parallel to the leading diagonal increase by 2
cThe differences from 49 are:
12 when 1 step away, 22 when 2 steps away, 32 when 3 steps away,…
The same pattern of differences occurs along each parallel diagonal through a square, for example,
4 × 6 = 52 − 12, 3 × 7 = 52 − 22, 2 × 8 = 52 − 32, 1 × 9 = 52 − 42
The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations.
The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.
© The University of Melbourne on behalf of the International Centre of Excellence for Education in
Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
How many six digit numbers can be formed using 3's and 5's only such that each number is a multiple of 6?
Detailed Solution n k ! Hence, the correct answer is 720.
How many 6
∴ In total, there are 900,000 6-digit numbers.
How many 3 digit numbers can be formed from the digits 2 3 5 6 and 7 which are divisible by 5 and in which none of the digit is repeated?
∴ Required number of numbers = (1 x 5 x 4) = 20.
How many possible 6
But with a six digit code, there are 1 million possible combos, making it a lot tougher for someone to crack your security code. If you are currently using a four digit PIN and update your software, you will need to manually opt in for the six digit PIN.