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Factors, multiples and primes

Factors and multiples are the same relationship read in two directions: 6 is a factor of 24, and 24 is a multiple of 6. Getting that into one idea rather than two saves a great deal of confusion later, because fractions, ratios and algebra all lean on it constantly.

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Finding every factor

How many factors does 24 have?

  1. Work up from 1, taking the pairs. 1 x 24, so 1 and 24.
  2. 2 x 12, so 2 and 12. 3 x 8, so 3 and 8. 4 x 6, so 4 and 6.
  3. 5 does not divide 24. Next would be 6, but 6 is already on the list — the pairs have met.
  4. The factors are 1, 2, 3, 4, 6, 8, 12 and 24. That is eight of them.

Worth knowing. Stopping when the pairs meet is what makes this quick. There is no need to test 7 through 23 — anything above the meeting point already turned up as the larger half of a pair.

The greatest common factor

What is the greatest common factor of 24 and 36?

  1. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
  2. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
  3. The ones in both lists: 1, 2, 3, 4, 6, 12.
  4. The greatest of those is 12.

Worth knowing. 6 is a common factor of 24 and 36, and it is a wrong answer. The question asks for the greatest one, and stopping at the first common factor found is the usual way to lose the mark.

Where it usually goes wrong

  • Giving a common factor rather than the greatest one.
  • Swapping factors and multiples — saying 24 is a factor of 6.
  • Calling 1 prime. It has only one factor, so it is neither prime nor composite.

The Professor finds factors in pairs. Every time he finds one he gets another free, and he knows to stop when the pair meets in the middle.

Practise it

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Questions teachers and parents ask

Is 1 a prime number?

No. A prime has exactly two factors, itself and 1, and 1 has only one factor because those are the same number. It is not composite either — it sits outside both groups, on purpose, because a great many rules in number theory would need an exception written into them otherwise.

What is the difference between a factor and a multiple?

A factor divides into a number; a multiple is built from it. 6 is a factor of 24, and 24 is a multiple of 6. It is one relationship read in two directions, which is why they are taught together.

When are these taught?

Factor pairs, multiples and prime versus composite for numbers to 100 are grade 4 (4.OA.B.4). Greatest common factor and least common multiple arrive in grade 6 (6.NS.B.4), where they are used to simplify fractions and to add fractions with different denominators.

What are the greatest common factor and the least common multiple actually for?

The greatest common factor puts a fraction in its simplest form: 24/36 divided top and bottom by 12 is 2/3. The least common multiple gives the denominator to use when adding fractions. Both are worth learning as tools rather than as tricks, because that is when they turn up again.

Where this shows up

Standards

  • 4.OA · Operations and algebraic thinking
  • 6.NS · The number system
  • CCSS.MATH.CONTENT.4.OA.B.4 — Find all factor pairs for a whole number 1-100, recognise multiples, and tell whether a number is prime or composite.
  • CCSS.MATH.CONTENT.6.NS.B.4 — Find the greatest common factor of two whole numbers up to 100 and the least common multiple of two whole numbers up to 12.

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