Factors, Multiples and Primes

This topic underpins fraction work, factoring and radicals, and it is the cheapest one on the subtest to get fully right, because it is almost all recognition.

Factors and multiples

Factors of 12: 1, 2, 3, 4, 6, 12. They divide into 12 with nothing left over.

Multiples of 12: 12, 24, 36, 48, and so on forever. They are what you get by multiplying 12 by whole numbers.

Every number is both a factor and a multiple of itself.

To list factors reliably, work in pairs from 1 upward and stop when the pair crosses over.

For 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6. That is nine factors, and you are done because the next try, 7, does not divide and 6 has already met itself. Listing in pairs is what keeps you from missing one.

Divisibility rules

Worth memorizing outright. Each takes a second and saves a division.

Is 4,158 divisible by 9?

Digits sum to 4 plus 1 plus 5 plus 8, which is 18, and 18 is a multiple of 9. So yes, and it is divisible by 3 and by 6 as well, since it is even.

There is no simple rule for 7, and the test knows that. If 7 is in question, just divide.

Primes

A prime number has exactly two distinct factors: 1 and itself.

The primes under 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Knowing that list cold answers a surprising number of questions outright.

Two facts the test asks directly:

1 is not prime, because it has only one factor, not two. It is not composite either; it is neither.

2 is the only even prime, because every other even number has 2 as a factor and so has at least three.

To test whether a number is prime, divide by the primes in order - 2, 3, 5, 7, 11 - and stop once the prime you are testing exceeds the square root. For 97, you need only test up to 9, so 2, 3, 5 and 7. None divide, so 97 is prime.

Prime factorization

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Every whole number above 1 breaks into primes exactly one way. Build it with a factor tree, splitting until every branch ends on a prime.

Factor 180.

180 is 18 times 10. Eighteen is 2 times 9, which is 2 times 3 times 3. Ten is 2 times 5.

So 180 is 2 x 2 x 3 x 3 x 5, or 2 squared times 3 squared times 5.

The starting split does not matter. Beginning with 4 times 45 gives the same primes, which is the point of the uniqueness.

Greatest common factor

The largest number that divides both. Used to reduce fractions and to split things into equal groups.

Two methods.

Listing, good for small numbers. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest shared is 12.

Prime factorization, better for larger numbers. Take the primes both share, each to the lower power.

• 24 is 2 cubed times 3.
• 36 is 2 squared times 3 squared.
• Shared: 2 to the lower power, which is squared, and 3 to the lower power, which is first. So 4 times 3 is 12.

Least common multiple

The smallest number both divide into. Used for common denominators and for questions about when two repeating events coincide.

Prime factorization method: take every prime that appears in either number, each to the higher power.

• 24 is 2 cubed times 3.
• 36 is 2 squared times 3 squared.
• Take 2 cubed and 3 squared: 8 times 9 is 72.

The shortcut worth knowing: GCF times LCM equals the product of the two numbers. Here 12 times 72 is 864, and 24 times 36 is 864. So once you have the GCF, the LCM is the product divided by it, which is usually faster than a second factorization.

That size row is the sanity check. A GCF bigger than the smaller number is wrong. An LCM smaller than the larger number is wrong.

What you can skip

Across the 25 questions on this topic:

• Divisibility rules for 7, 11 and 13. The divisibility questions stay with 2, 3, 4, 5 and 9.
• Perfect numbers and other number-theory curiosities never appear. "Perfect" appears only as "perfect square".

Where people lose points

Calling 1 prime. It is not.

Forgetting 2 is prime because it is even.

Swapping GCF and LCM. The size check catches it instantly.

Taking the higher power for the GCF or the lower for the LCM.

Missing a factor when listing. Work in pairs.

Confusing factors with multiples. Factors are at most the number; multiples are at least the number.

Work one in under a minute

Two lights flash on different cycles, one every 12 seconds and one every 18. They flash together now. When do they next flash together?

Two cycles meeting again is an LCM question.

• 12 is 2 squared times 3.
• 18 is 2 times 3 squared.
• Higher powers: 2 squared times 3 squared is 4 times 9, which is 36 seconds.

Check: 36 is a multiple of both, and it is at least as large as 18. It is.

Where this leads

Prime factorization is how radicals get simplified and how fractions get reduced, and the LCM is the common denominator you need for every fraction sum.