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Factors & primes

lesson · about 3 minutes

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

Best in your head · no pen, no calculator why?

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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.

Estimate the square root, then try each prime up to it.
Shortcuts: 2 divides even numbers, 5 divides endings of 0 or 5, and 3 divides when the digit sum does.
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?

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

Answer: 7

Split into primes

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

Divide by the smallest prime that works, and write it down.
Repeat until what is left is prime.
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?

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

Answer: 11

Shared primes and all primes

The GCD or LCM of two numbers.

Factor both numbers into primes.
GCD: multiply the primes they share, each at the lower power.
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?

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

Answer: 60

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.

practice

Spot the prime

worked example

Which number is prime?

Answer: 71

Test the primes up to √71: 2, 3, 5 and 7 (9² = 81 is past 71). None divides 71, so it is prime.
The others: 93 = 3 × 31, 129 = 3 × 43, 133 = 7 × 19.

Largest prime factor

worked example

What is the largest prime factor of 34?

Answer: 17

34 = 2 × 17.
The largest prime factor is 17.

Greatest common divisor

worked example

What is the greatest common divisor of 198 and 154?

Answer: 22

198 = 2 × 3² × 11 and 154 = 2 × 7 × 11.
Take the primes they share, each at its lower power: 2 × 11 = 22.
Check: 198 = 22 × 9 and 154 = 22 × 7, and 9 and 7 share no factor other than 1.

Least common multiple

worked example

What is the least common multiple of 20 and 26?

Answer: 260

20 = 2² × 5 and 26 = 2 × 13.
Take every prime at its higher power: 2² × 5 × 13 = 260.
Check: 20 × 26 ÷ 2 (their greatest common divisor) = 520 ÷ 2 = 260.