The Factor Counter

Questions about 'number of factors,' 'GCF,' 'LCM,' 'is X a factor of Y,' or anything involving factorials.

Trying to list factors one by one. Wastes time. Also: forgetting that for 'number of factors,' you need (a+1)(b+1)(c+1) — not just a·b·c.

1. Prime factorize the number completely. Example: 60 = 2² · 3 · 5.
2. For 'number of factors': if N = p^a · q^b · r^c, then number of factors = (a+1)(b+1)(c+1). For 60: (2+1)(1+1)(1+1) = 12.
3. For 'is X a factor of Y': check that every prime in X's factorization appears in Y's with at least equal power.
4. For GCF (greatest common factor): take the MINIMUM power of each shared prime.
5. For LCM (least common multiple): take the MAXIMUM power of each prime across both numbers.
6. For factorials: count prime powers using Legendre's formula — the power of prime p in n! is floor(n/p) + floor(n/p²) + …

Factorization is the universal solvent for divisibility questions. Master this one pattern and roughly 15% of hard Quant becomes mechanical.