Factors, Multiples & Primes
Grades 4–8 Number theory
Factors are the numbers that divide evenly into a number — factors of 12: 1, 2, 3, 4, 6, 12. Multiples are what you get by multiplying — multiples of 12: 12, 24, 36, 48… Factors multiply up to the number; the number multiplies out to its multiples. Same idea, opposite directions.
Divisibility tricks worth memorizing
- 2: ends in an even digit
- 3: digit sum divides by 3 (741 → 7+4+1 = 12 → yes)
- 4: last two digits divide by 4
- 5: ends in 0 or 5
- 9: digit sum divides by 9
- 10: ends in 0
Primes — the atoms of arithmetic
A prime has exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19… Every whole number is built from a unique recipe of primes — the Fundamental Theorem of Arithmetic:
Factor trees find the recipe: keep splitting any composite branch until every leaf is prime. Order doesn't matter — 360's recipe is the same no matter how you break it down.
How to test a number for primality
You only need to check prime factors up to the square root — because if n = a×b and both a,b > √n, then a×b > n. To test 97: √97 ≈ 9.8, so check 2, 3, 5, 7 — none divide it → 97 is prime.
- Rivest, R. L., Shamir, A., & Adleman, L. (1978). A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2), 120–126. doi:10.1145/359340.359342 — the original RSA paper, in the ACM Digital Library.
- Agrawal, M., Kayal, N., & Saxena, N. (2004). PRIMES is in P. Annals of Mathematics, 160(2), 781–793. doi:10.4007/annals.2004.160.781 — the breakthrough proof that primality can be tested quickly, settling a centuries-old open problem.
- Is 91 prime? (no — 7 × 13)
- Prime-factor 84 (2² × 3 × 7)
- Is 101 prime? (check 2,3,5,7 → yes)
Next: see primes and binary working inside real machines on the math in computing page, or division for the operation behind it all.