Factors

1.0 Numbers · 1.2 Factors

Syllabus tag: Kenya CBC | Grade 7 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.2 Factors (7 lessons)

Lesson objectives

By the end of this topic, you will be able to:

Test divisibility of numbers by 2, 3, 4, 5, 6, 8, 9, 10 and 11
Express composite numbers as a product of prime factors
Work out the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of numbers by the factor method
Apply the GCD and LCM in real-life situations

A factor of a number divides into it exactly, leaving no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12.

a) Divisibility tests

These let you spot factors without dividing.

Divisible by	Test
2	the last digit is even
3	the digits add to a multiple of 3
4	the last two digits form a multiple of 4
5	it ends in 0 or 5
6	it passes both the 2 test and the 3 test
8	the last three digits form a multiple of 8
9	the digits add to a multiple of 9
10	it ends in 0
11	alternate digits added, then subtracted, gives 0 or a multiple of 11

Take 468. It is even, so 2 divides it. Its digits add to 18, a multiple of 9, so 9 divides it too. Passing both 2 and 3 means 6 divides it.

b) Prime and composite

A prime number has exactly two factors: 1 and itself. The first few are 2, 3, 5, 7, 11 and 13.

A composite number has more than two factors.

The number 1 is neither. It has only one factor, so it fails the definition of prime.

The number 2 is the only even prime, because every other even number has 2 as a factor.

c) Prime factorisation

Every composite number can be written as a product of primes, in exactly one way.

Divide by the smallest prime that goes in, then keep dividing until everything left is prime.

Write the answer in index form to keep it short.

d) GCD and LCM from the factors

The Greatest Common Divisor is the largest number dividing into both.

The Least Common Multiple is the smallest number both divide into.

For the GCD, take each prime that appears in both, at its lowest power.

For the LCM, take every prime that appears, at its highest power.

Those two rules are opposites, and mixing them is the usual error. GCD is small and LCM is large, so check your answers are the right size.

e) A useful check

For any two numbers, GCD × LCM equals the product of the numbers.

Here 12 × 144 = 1 728, and 36 × 48 = 1 728. They match, so both answers are right.

f) Where this is used

GCD shares things into the largest equal groups. Think of cutting cloth into the longest equal pieces with none left over.

LCM finds when things happening at different intervals coincide, such as two buses leaving together again.

Words to know

Divisibility test — a shortcut method for checking whether one number divides exactly by another.
Composite number — a whole number with more than two factors.
Prime factorisation — expressing a composite number as a product of its prime factors.
Greatest Common Divisor (GCD) — the largest number that divides two or more numbers exactly.
Least Common Multiple (LCM) — the smallest number that two or more numbers divide into exactly.

:::checkpoint Check yourself

Try the exercises below — on divisibility tests, prime factorisation, GCD and LCM.