Factors and Multiples

We want to find numbers which exactly divide 4. To do that we start rom 1 and divide 4 by numbers less than 4 as below.

1 ) 4 ( 4
-4
0

Quotient is 4
Remainder is

4 = 1 × 4

2 ) 4 ( 2
-4
0

Quotient is 2
Remainder is

4 = 2 × 2

3 ) 4 ( 1
-3
1

Quotient is 1
Remainder is

4 ) 4 ( 1
-4
0

Quotient is 1
Remainder is

4 = 4 × 1

As you can see, some numbers divide 4 exactly with 0 reminder. We find that the number 4 can be written as: 4 = 1 × 4; 4 = 2 × 2; 4 = 4 × 1 and know that the numbers 1, 2 and 4 are exact divisors of 4. These numbers are called factors of 4.

Observe each of the factors of 4 is less than or equal to 4.

4	×	5	=	20
factor	×	factor	=	multiple

We can say that a number is a multiple of each of its factors

We have provided a divison calculator below. Take a any number for the dividend .Find the numbers below the selected number which divide it exactly and leave a reminder 0. Start with 1 and increase and note down your observations.

By now you should be comfortable with addition, subtraction and multiplication of integers. Division is slightly different, because you can’t always divide any integer by any other. For example 17 divided by 3 is not a whole number – it is somewhere in between 5 and 6. You either have to give a remainder (2), or express the answer as a decimal number (5.66…).

12 is divisible by 3

10 is not divisible by 4

If you can divide a number A by a number B, without remainder, we say that B is a factor (or divisor) of A, and that A is a multiple of B. We often write B/A, where the slanting line simply means “divides”.

For example, 7 × 3 = 21, so 7 is a of 21. Similarly, 21 is a of 7, and we can write 7/21.

In this short game you have to determine which numbers are factors or multiples:

Factors and Multiples Quiz

${x}

is a

factor

multiple

neither

${y}

It is often useful to find all the factors of a number. For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Of course, you don’t want to check all numbers up to 60 if they are factors. Instead, there is a simple technique which relies on the fact that factors always appear in .

In the case of 60 we have 60 = 1 × 60 = 2 × 30 = 3 × 20 = 4 × 15 = 5 × 12 = 6 × 10. Or, in a different notation,

60	1,	2,	3,	4,	5,	6,	10,	12,	15,	20,	30,	60

To find all factors of a number we simply start at both ends of this list, until we meet in the middle.

42	1,	2,	3,	6,	7,	14,	21,	42

For example, the first factor pair of 42 is simply 1 and 42, and we write them down with much space in between.

After 1 at the beginning, we check if 2 divides 42. It does, and the corresponding pair is 42 ÷ 2 = 21.

Next, we check if 3 divides 42. It does, and the corresponding pair if 42 ÷ 3 = 14.

Now we check if 4 divides 42. It does not, however, so we move on.

5 also doesn’t divide 42 so we move on.

6 does divide 42 again. Its pair is 42 ÷ 6 = 7. Notice how we’ve met in the middle after only a few attempts, without having to test all numbers from 7 to 42.

The only special case with this method is for square numbers: in that case, you will meet at just a single number in the middle, like 64 = × .

Try these

1. Find the possible factors of 45, 30 and 36.

Factors of 45: , , , , and

Factors of 30: , , , , , , and

Factors of 36: , , , , , , , and

Factors of 68: , , , , and

2. Write first five multiples of 6.

The required multiples are: 6×1= , 6×2 = , 6×3 = , 6×4 = , 6×5 =

i.e. 6, 12, 18, 24 and 30.

Let us see what we conclude about factors :

Instructions

Is there any number which occurs as a factor of every number . It is . For example 6 = 1 × 6, 18 = 1 × 18 and so on.Check it for a few more numbers.

We say 1 is a factor of every number.

Can 7 be a factor of itself. You can write 7 as 7 = × 1. What about 10? and 15?.You will find that every number can be expressed in this way.

We say that every number is a factor of itself.

What are the factors of 16? They are 1, 2, 4, 8, 16. Out of these factors do you find any factor which does not divide 16 . Try it for 20; 36.

You will find that every factor of a number is an exact divisor of that number.

What are the factors of 34? They are 1, 2, and itself. Out of these which is the greatest factor.

The other factors 1, 2 and 17 are less than 34. Try to check this for 64,81 and 56.

We say that every factor is less than or equal to the given number.

The number 76 has 5 factors. How many factors does 136 or 96 have? You will find that you are able to count the number of factors of each of these. Even if the numbers are as large as 10576, 25642 etc. or larger, you can still count the number of factors of such numbers, (though you may find it difficult to factorise such numbers).

We say that number of factors of a given number are .

Let us see what we conclude about multiples:

Instructions

What are the multiples of 7? (write in order) 7, 14, , , ,... You will find that each of these multiples is greater than or equal to 7. Will it happen with each number? Check this for the multiples of 6, 9 and 10.

We find that every multiple of a number is greater than or equal to that number.

Write the multiples of 5. They are 5, 10, , , , ... Do you think this list will end anywhere? No! The list is endless. Try it with multiples of 6,7 etc.

We find that the number of multiples of a given number is infinite.

Can 7 be a multiple of itself ? Yes, because 7 = 7 × 1. Will it be true for other numbers also? Try it with 3, 12 and 16.

You will find that every number is a of itself.

The factors of 6 are 1, 2, and . Also, 1 + 2 + 3 + 6 = 12 = 2 × .

All the factors of 28 are 1, 2, , , and .

Adding these we have, 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × .

The sum of the factors of 28 is to twice the number 28.

So the numbers 6 and 28 are perfect numbers.

Is 10 a perfect number

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Factors and Multiples

Factors and Multiples Quiz

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Factors and Multiples Quiz

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