Factors and Multiples Lessons: All 13 in Order

Dividing Exactly #

Sometimes the sharing leaves nothing behind.

One number divides another exactly when the sharing leaves nothing behind

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

11 ÷ 5 has a remainder of 1.

Now you

Share 28 into rows of 3. How many are left over?

Share 36 into rows of 4. How many are left over?

Factors #

The numbers that divide in with nothing left over.

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

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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.

the factors of 12

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

Now you

the factors of 20

Which of these is a factor of 20?

the factors of 30

Which of these is a factor of 30?

Factor Pairs #

They arrive two at a time.

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

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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.

Now you

How many factors does 12 have?

18 = 6 × what?

Multiples #

What you land on counting up in a number.

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

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multiples of 4

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.

Now you

multiples of 7

Which is a multiple of 7?

multiples of 6

Which is a multiple of 6?

Divisibility Tests #

Tell that it divides without doing the division.

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

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even numbers

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

ending 0 or 5

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.

Now you

A number divides by 10…

Which of these divides by 2?

The Sieve of Eratosthenes #

Cross out every multiple and the primes remain.

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

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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.

Now you

multiples of 7

Crossing out multiples of 7 removes which of these?

multiples of 2

Crossing out multiples of 2 removes which of these?

Prime Factorization #

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

Every number has exactly one prime factorization

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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.

Now you

Break 30 into primes

Break 8 into primes

Common Factors #

The factors two numbers happen to share.

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

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factors of 12

These six numbers divide 12 exactly.

factors of 18

And these six divide 18.

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

Now you

Which divides both 9 and 12?

Which divides both 8 and 12?

The Highest Common Factor #

The biggest one they both own.

The highest common factor is the largest number that divides both

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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.

Now you

8 = 2 × 2 × 2 and 12 = 2 × 2 × 3. Their HCF?

Highest common factor of 10 and 15?

Common Multiples #

Where two counts land on the same number.

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

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multiples of 4

These are the multiples of 4.

multiples of 6

And these are the multiples of 6.

both land here

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

LCM = 12

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

Now you

multiples of 4

Which do both 4 and 6 divide into?

multiples of 4

Which do both 4 and 10 divide into?

The Lowest Common Multiple #

The first time two counts meet.

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

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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.

Now you

Lowest common multiple of 4 and 10?

18 = 2 × 3 × 3 and 30 = 2 × 3 × 5. Their LCM?

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

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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.

Now you

Lighthouses flashing every 9 and every 12 seconds. When do they agree?

Ropes of 16 m and 24 m cut into equal lengths. Longest piece?

HCF and LCM Word Problems #

Choose the tool from the story, then compute it.

Choose HCF or LCM from the story, then compute it

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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.

Now you

Ribbons 16 cm and 24 cm: longest equal piece, no waste?

Buses every 9 and every 12 min leave together now. Next time?