doing math

courses › Pre-Algebra

Factors & primes

level 14 course

Primes, prime factors, greatest common divisor, least common multiple.

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Opens at level 14.

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Builds on: Division (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A prime is a whole number above 1 whose only divisors are 1 and itself: 2, 3, 5, 7, 11, 13 and so on. Every other whole number above 1 splits into primes in only one way, apart from order: its prime factorization. 84 is 2 × 2 × 3 × 7.

That split answers most questions here, including the greatest common divisor (GCD), the biggest number that divides both of two numbers, and the least common multiple (LCM), the smallest number both divide into.

Techniques

Test primes to the square root

Deciding whether a number is prime.

  1. Estimate the square root, then try each prime up to it.
  2. Shortcuts: 2 divides even numbers, 5 divides endings of 0 or 5, and 3 divides when the digit sum does.
  3. If none divides it, it is prime: any bigger factor would pair with a smaller one you tried.
worked example

Example: What is the smallest prime factor of 91?

  1. √91 is between 9 and 10, so try 2, 3, 5 and 7.
  2. 91 is odd, its digits sum to 10, and it ends in 1.
  3. 91 ÷ 7 = 13.

Answer: 7

Split into primes

Largest prime factor, and the first step for GCD and LCM.

  1. Divide by the smallest prime that works, and write it down.
  2. Repeat until what is left is prime.
  3. Those primes and the one left over are the factorization. The leftover prime is the largest.
worked example

Example: What is the largest prime factor of 132?

  1. 132 ÷ 2 = 66, and 66 ÷ 2 = 33.
  2. 33 ÷ 3 = 11, and 11 is prime.
  3. 132 = 2 × 2 × 3 × 11.

Answer: 11

Shared primes and all primes

The GCD or LCM of two numbers.

  1. Factor both numbers into primes.
  2. GCD: multiply the primes they share, each at the lower power.
  3. LCM: multiply every prime that appears in either, each at the higher power.
worked example

Example: What is the least common multiple of 20 and 30?

  1. 20 is 2² × 5, and 30 is 2 × 3 × 5.
  2. Every prime at its higher power: 2² × 3 × 5 = 60.
  3. Check: the GCD is 2 × 5 = 10, and 10 × 60 = 600 = 20 × 30.

Answer: 60

Tips by skill

  • TipSpot the prime: Try each prime up to the square root. Numbers ending in 1, 3, 7 or 9 can still split, like 7 × 13.
  • TipLargest prime factor: Keep dividing by the smallest prime that works until what is left is prime. That last prime is the largest.
  • TipGreatest common divisor: Factor both numbers. Multiply only the primes they share, each at the lower power.
  • TipLeast common multiple: Factor both. Take every prime at its higher power, or multiply the numbers and divide by their GCD.

Watch out for

  • Calling 91 prime. It is odd, and neither 3 nor 5 divides it, but it is 7 × 13.
  • Giving the largest factor instead of the largest prime factor. 42 divides 84, but 42 is not prime.
  • Multiplying the two numbers for the LCM. 12 × 18 is a common multiple; divide by their GCD, 6, for the least.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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