Factors and Multiples

Dividing Exactly #

Sometimes the sharing leaves nothing behind.

One number divides another exactly when the sharing leaves nothing behind

12 squares can be arranged into 4 rows of 3, with none left over.

11 ÷ 5 has a remainder of 1.

Factors #

The numbers that divide in with nothing left over.

A factor of a number is any number that divides it exactly

12 counters make a full rectangle 2 rows by 6, so 2 and 6 both divide 12.

They also make 3 by 4.

And one long row: 1 × 12, so 1 and 12 divide 12 as well.

12 has six factors: 1, 2, 3, 4, 6 and 12. Nothing else divides it exactly.

Factor Pairs #

They arrive two at a time.

Factors come in pairs, which is what lets you find every one of them

Every rectangle of 18 names a pair: 1 × 18 = 2 × 9 = 3 × 6 — six factors in all.

Walk up from 1: every number that divides names its partner. Stop where they cross.

Multiples #

What you land on counting up in a number.

The multiples of a number are what you land on counting up in it

Count up in fours and you land on 4, 8, 12, 16 — the multiples of 4.

Each one is a whole number of jumps of 4 away from zero. They never stop.

Divisibility Tests #

Tell that it divides without doing the division.

A few quick checks tell you a number divides without doing the division

A number divides by 2 when its last digit is even.

It divides by 5 when it ends in 0 or 5, and by 10 only when it ends in 0.

To test for 3, add all the digits. If that total divides by 3, so does the number.

The Sieve of Eratosthenes #

Cross out every multiple and the primes remain.

The Sieve of Eratosthenes: crossing out every multiple leaves just the primes

Write out 1 to 100. A prime has exactly two factors: 1 and itself.

Cross out 1 first: its only factor is 1, so it has one factor, not two.

Cross out multiples of 2, then 3, then on. Whatever survives the sieve is prime.

Prime Factorization #

Every number is a product of primes in one way only.

Every number has exactly one prime factorization

Split 24 into 4 × 6 and keep going until only primes are left.

Split 24 as 3 × 8 and the same primes appear — the Fundamental Theorem of Arithmetic.

Common Factors #

The factors two numbers happen to share.

A common factor is a number that divides both of two numbers

These six numbers divide 12 exactly.

And these six divide 18.

The ones in the overlap divide both. Those are the common factors.

The Highest Common Factor #

The biggest one they both own.

The highest common factor is the largest number that divides both

12 and 18 share 1, 2, 3 and 6. The largest is 6, the Highest Common Factor (HCF).

From primes: multiply what both lists carry — a 2 and a 3 — and 2 × 3 = 6.

Common Multiples #

Where two counts land on the same number.

A common multiple is a number that both of two numbers divide into

These are the multiples of 4.

And these are the multiples of 6.

Both counts land on 12 and on 24. Those are their common multiples.

12 is the Lowest Common Multiple (LCM) of 4 and 6.

The Lowest Common Multiple #

The first time two counts meet.

The lowest common multiple is the first number both counts land on

Counting in fours: 4, 8, 12, 16, 20, 24.

Counting in sixes: 6, 12, 18, 24. The first meeting point is 12.

The lowest common multiple — LCM — is 12, not 24: the earliest meeting.

From primes: take each prime the most times either list uses it — 2 × 2 × 3 = 12.

Choosing Between HCF and LCM #

The HCF divides two numbers. The LCM is what they divide into.

The answer dividing the numbers needs the HCF, and the numbers dividing the answer needs the LCM

Cutting two ribbons of 12 and 18 into equal pieces: the longest is the HCF, 6.

Two buses leaving every 4 and every 6 minutes meet again at the LCM, 12.

HCF divides the two numbers. LCM is what they both divide into.

HCF gave 6, under both 12 and 18. LCM gave 12, over both 4 and 6.

HCF and LCM Word Problems #

Choose the tool from the story, then compute it.

Choose HCF or LCM from the story, then compute it

Strips 27 cm and 36 cm, cut equally with no waste: the HCF. Shared primes: 9.

Bells every 6 and every 10 minutes ring together at the LCM: 30 minutes.