A worked example: 60's prime factorization and full factor list
The Prime Factorization Calculator breaks 60 down into 2^2 × 3 × 5, with distinct primes 2, 3, and 5. Feed that same number, 60, into the Factor Calculator and it returns every divisor: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 — twelve factors in total. Every single one of those twelve numbers can be built by multiplying some combination of the primes 2, 3, and 5, using each prime up to the power it appears in the factorization.
The formula: multiplying subsets of prime powers
Since 60 = 2^2 × 3 × 5, the exponent on 2 can be chosen as 0, 1, or 2 (three choices), while the exponents on 3 and 5 can each only be 0 or 1 (two choices apiece). Multiplying the choices together, 3 × 2 × 2 = 12, gives exactly the factor count the calculator reported. The factors themselves fall straight out of the combinations: 2^0 × 3^0 × 5^0 = 1, 2^1 = 2, 2^2 = 4, 3^1 = 3, 5^1 = 5, 2 × 3 = 6, 2 × 5 = 10, 2^2 × 3 = 12, 3 × 5 = 15, 2^2 × 5 = 20, 2 × 3 × 5 = 30, and 2^2 × 3 × 5 = 60 — all twelve appear in the calculator's list, and nothing else does.
Reconstructing factors for a second number: 24
The same chain works for any number. The Prime Factorization Calculator breaks 24 down into 2^3 × 3, with distinct primes 2 and 3. That gives an exponent choice of 0 through 3 for 2 (four choices) and 0 or 1 for 3 (two choices), so 4 × 2 = 8 combinations — matching the eight factors the Factor Calculator actually returns for 24: 1, 2, 3, 4, 6, 8, 12, 24. Checking the combinations directly: 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8, 3^1 = 3, 2 × 3 = 6, 2^2 × 3 = 12, and 2^3 × 3 = 24 — again, every factor in the list is accounted for.
Why this connection matters
This is why prime factorization is often called the "DNA" of a number — once you know it, you don't need to test every candidate divisor up to the square root to find the full factor list; you can generate it directly by combining prime powers. It also explains why the two calculators' results always agree: 60's twelve factors and 24's eight factors are not independent facts, they are the direct, countable consequence of each number's prime factorization.