About Factors
Factoring is arithmetic's disassembly manual: homework (simplifying fractions, factoring quadratics), practical splitting (dividing 60 students into equal teams — the factor list IS the option list), and number theory's favorite pastime (primes, perfect numbers) all start from the same list this tool produces.
Enter any whole number up to 10 million for the complete factor list, count, prime factorization, and factor pairs. Two-factor results get the prime badge; the pairs view doubles as a 'ways to arrange in a rectangle' answer for everything from classroom seating to tile layouts.
Comparing factors ACROSS two numbers? That's the GCF Calculator.
The Square-Root Method
Why nobody checks all the way up to n:
For i from 1 to √n: if n mod i = 0 → i and n÷i are both factors Factor count from primes: (e₁+1)(e₂+1)… for n = p₁^e₁ × p₂^e₂…
Worked example: 60 needs checks only to 7 (√60 ≈ 7.7), yielding pairs 1×60, 2×30, 3×20, 4×15, 5×12, 6×10 — twelve factors, matching the exponent formula on 2²×3×5: (2+1)(1+1)(1+1) = 12. ✓
Worked Examples
Numbers people actually look up — computed by this calculator:
| Number | Factors | Prime factorization | Count |
|---|---|---|---|
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 2³ × 3 | 8 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 2² × 3² | 9 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 2² × 3 × 5 | 12 |
| 97 | 1, 97 | 97 (prime) | 2 |
| 144 | 15 factors, 1 through 144 | 2⁴ × 3² | 15 |
Odd factor counts (36's nine, 144's fifteen) mark perfect squares — their square root pairs with itself, breaking the even-pairs pattern.
Divisibility Shortcuts
The classics that speed hand-factoring: 2 (even last digit), 3 (digit sum divisible by 3), 4 (last two digits divisible by 4), 5 (ends in 0/5), 6 (rules for 2 and 3 both), 9 (digit sum divisible by 9), 10 (ends in 0). For 11: alternately add and subtract digits — divisible if the result is. Combined, they crack most everyday numbers before a calculator wakes up.
Prime factorization's quiet superpowers: any factor of n is a product of its prime factors taken with equal-or-smaller exponents (which is how the count formula works), GCF takes the shared primes at minimum exponents while LCM takes all primes at maximum, and cryptography's foundation is precisely that factoring VERY large numbers stays hard — your number here surrenders in microseconds; a 600-digit one guards bank transfers.