Factors, Multiples, and Prime Factorization

Prime factorization writes a number as a product of primes, like 360 = 2³ times 3² times 5. This decomposition is unique: no other set of primes multiplies to 360. That uniqueness is why primes work like atoms for numbers. A factor tree finds it by repeatedly splitting a number into any two factors until every branch ends in a prime: 360 = 36 times 10 = (6 times 6) times (2 times 5) = (2 times 3 times 2 times 3) times (2 times 5), which collects into 2³ times 3² times 5.

Once you have the prime factorization, counting factors becomes arithmetic instead of listing. For N = 2³ times 3², each prime's exponent tells you how many powers of that prime can appear in a factor: 0 through 3 for the 2 (four choices), 0 through 2 for the 3 (three choices). Multiply the choices: 4 times 3 = 12 factors. In general, if N = p^a times q^b times r^c for distinct primes, the factor count is (a+1)(b+1)(c+1).

This connects directly to squaring: if N = 2^a times 3^b, then N² = 2^2a times 3^2b, so every exponent doubles and the factor count becomes (2a+1)(2b+1), always odd. That oddness is a signature of perfect squares. The same exponent logic also explains why 1 cannot be prime: a prime needs exactly two distinct divisors, but 1 has only one, itself, so letting it into a factorization would wreck the one-way-only guarantee that makes this whole counting method work.