🔢 Number Kingdom · Number Theory
Factors, Multiples & Primes
Take numbers apart into their factors and build them up as multiples. Primes are the atoms every whole number is made from.
In short
- Factors divide into a number; multiples are what you count up to. Factors are never larger than the number, multiples never smaller.
- A prime has exactly two factors; 1 is not prime and 2 is the only even prime.
- Every number above 1 has exactly one prime factorisation.
- GCF uses shared primes at the lower power; LCM uses all primes at the higher power.
Factors go down, multiples go up
A factor of a number divides into it exactly, with nothing left over. A multiple is what you get by counting up in that number.
For 12:
- factors: 1, 2, 3, 4, 6, 12 — all of them are 12 or smaller
- multiples: 12, 24, 36, 48, 60, ... — all of them are 12 or larger
The two words are easy to swap by accident, so anchor them: factors are the pieces a number breaks into; multiples are the places you land when you count in that number.
Finding every factor
Factors come in pairs. For 36: 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6. Listing the pairs in order guarantees you miss none.
You only have to test divisors up to the square root of the number. Past that point the pairs start repeating in reverse. For 36 the square root is 6, so testing 1 through 6 finds everything.
A number with exactly one pair that repeats (like 6 x 6) is a square number, and it has an odd number of factors — 36 has nine.
Primes are the atoms
A prime has exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19, 23, ... A composite number has more than two.
Two edge cases matter:
- 1 is not prime. It has only one factor, not two.
- 2 is prime, and it is the only even prime, because every other even number has 2 as an extra factor.
To test a number, divide by the primes in order — 2, 3, 5, 7, 11 — and stop once the prime you are testing passes the square root. Quick checks: even numbers above 2 are composite; a number whose digits add to a multiple of 3 is divisible by 3; a number ending in 0 or 5 is divisible by 5.
Prime factorisation
Every whole number above 1 can be written as a product of primes in exactly one way. That is the Fundamental Theorem of Arithmetic, and it is why primes are called the atoms of arithmetic.
Build one with a factor tree: split the number into any two factors, keep splitting the composite branches, and stop when everything is prime.
- 60 = 6 x 10 = (2 x 3) x (2 x 5) = 22 x 3 x 5
Starting with 4 x 15 instead of 6 x 10 gives the same primes in the end. The route does not matter; the destination is unique.
GCF and LCM
The greatest common factor (GCF) of two numbers is the largest number that divides into both. The least common multiple (LCM) is the smallest number both divide into.
Prime factorisations make both easy:
24 = 23 x 3 36 = 22 x 32
- GCF: take the primes they share, each to its smaller power: 22 x 3 = 12.
- LCM: take every prime that appears, each to its larger power: 23 x 32 = 72.
Two sanity checks that catch a swap immediately: the GCF is never larger than the smaller number, and the LCM is never smaller than the larger number. There is also a shortcut: GCF x LCM = the product of the two numbers (12 x 72 = 24 x 36).
Worked examples
Example 1
Find the GCF and the LCM of 18 and 30.
- Prime factorise: 18 = 2 x 32 and 30 = 2 x 3 x 5.
- Shared primes with the smaller power: 21 and 31, so GCF = 2 x 3 = 6.
- All primes with the larger power: 21, 32, 51, so LCM = 2 x 9 x 5 = 90.
- Check the sizes: 6 is not larger than 18, and 90 is not smaller than 30.
- Check the shortcut: 6 x 90 = 540 = 18 x 30.
Example 2
Is 91 prime or composite?
- 91 is odd, so 2 does not divide it.
- Its digits add to 9 + 1 = 10, which is not a multiple of 3, so 3 does not divide it.
- It does not end in 0 or 5, so 5 does not divide it.
- Test 7: 7 x 13 = 91, exactly.
- 91 has the factors 1, 7, 13 and 91, so it is composite — not prime, despite looking like one.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 5How many factors does 18 have? (Count 1 and 18 as well.)
Answer: 6
- Factor pairs of 18: 1 x 18, 2 x 9, 3 x 6
- All factors: 1, 2, 3, 6, 9, 18
- That is 6 factors.
Problem 2
Difficulty 3 of 5What is the greatest common factor (GCF) of 16 and 104?
Answer: 8
- 16 = 24
- 104 = 23 x 13
- The primes they share multiply to 8.
- GCF(16, 104) = 8
Problem 3
Difficulty 4 of 5What is the least common multiple (LCM) of 19 and 23?
Answer: 437
- Multiples of 19: 19, 38, 57, 76, 95, ...
- Multiples of 23: 23, 46, 69, 92, 115, ...
- GCF(19, 23) = 1, so LCM = 19 x 23 / 1 = 437
Common mistakes
- Swapping GCF and LCM — giving a "common factor" that is larger than both numbers.
- Calling 1 prime, or calling 2 composite because it is even.
- Missing factor pairs by testing divisors unsystematically.
- Deciding a number is prime after testing only 2, 3 and 5, when 7 or 11 still divides it.
What you should be able to do
- List all factors of a number and decide whether it is prime.
- Write the prime factorisation of a number.
- Find the greatest common factor of two numbers.
- Find the least common multiple of two numbers.
Where this fits in the curriculum
Common Core
- 4.OA.B.4
Grade 4 — Find all factor pairs of a whole number and decide whether it is prime or composite.
- 6.NS.B.4
Grade 6 — Find the greatest common factor and the least common multiple of two whole numbers.
- 6.EE.A.2.B
Grade 6 — Identify the parts of an expression, including viewing a product as a single entity.
Prime factorisation itself is never named as a Common Core standard; it is the technique behind 6.NS.B.4 rather than a standard of its own.