Unit 3 sec 1.2

Unit 3 sec 1.2 Factors

04 November 2015

23:27

Natural number that divides exactly into a second natural number is called a factor or devisor all of the second number

 factors are closely related to multiples, science ‘2 is a factor of 10’means the same as ‘10 is a multiple of 2’. Every number greater than 1 has at least 2 factors, itself, and 1, but most numbers have more factors than this.

 The factor of a number can be arranged into factor pairs, where the 2 factors in each pair multiple together to give the number. For example, the factor pair of 10 are

1, 10 (since 1×10 =10),

2, 5 (since 2×5=10).

You can use the idea of factor pairs to help you find all the factors of a number.

 Strategy to find a factor of a number

v try the numbers 1, 2, 3, 4, in turn. Whenever you find a factor, write down the other factor of the factor pair.

v So when you get a factor pair that you have already.

Example 1, finding the factor of a number

find all the factors of 28

solution

the 1^st factor pair of the number is always 1 and the number. 1^st factor pair is 1, 28.

Try 2.

The next factor pair is 2, 14.

Try 3: it is not factor. Try 4

the next factor pair is 4, 7.

Try 5: it is not a factor. Try 6: it is not a factor. Try 7: this is a factor pair 7, 4, which is already found, so stop. To finish, but that is in increasing order.

The factor is 28 are 1, 2, 4, 7, 14 and 28.

 Because the factors of a number, form pairs, most numbers have an even number of factors. The only exceptions are the square numbers, each of which has an ordered number of factors. This is because one of the factors of a square number pairs with itself.

The test in the box below can be useful when you are trying to find the factors of a number. They give you a quick way to tell whether a given number is divisible by 2,3, 5 or 9.

Divisibility tests

a number is divisible by

ª    2. If it ends in 0, 2, 4, 6 or 8

ª    3. If it digits at up to a multiple of 3

ª    5. If it ends in 0 or 5

ª    9. If its digits and up to a multiple of 9.

If the number does not satisfy a test above, then it is not advisable by a specified number.

Common factors

the numbers that appear in both lists are called the common factors of 12 and 18. So the common factors of 12 and 18 are 1, 2, 3, and 6. The largest common factor of 12 and 18 is 6, and this is usually called the highest common factor. These ideas are summarised in the box below.

A common factor of two or more numbers is a number that is a factor of all of them. The highest common factor (HCF) of two or more numbers is the highest number that is a factor of all of them. An alternative name of the highest common factor is greatest common devisor (GCD).

Is common factors can be useful when you are cancelling fractions. When you want to cancel a fraction down to its simplest form, you need to divide top and bottom by the highest common factor of the new numerator and denominator. The numerator and denominator of the simplified fraction have no common factors, other than 1, so the fraction is in its simplest form.

It does not matter if you did not immediately spot the highest common factor of the numerator or and denominator of a fraction - you can always cancel the fraction down in stages.

   There is a quicker way to find highest common factors than the method that you have seen in this subsection. It involves prime numbers, which you will learn about in next subsection.