Divisor Count from Prime Exponents

You do not have to list the divisors of a number to count them. Factor it, add one to each prime exponent, multiply the results. For 360 that is $2^3 \cdot 3^2 \cdot 5^1$, giving $4 \times 3 \times 2 = 24$ divisors — and there really are 24, from 1 up to 360 itself. The reason is a counting argument rather than an algebraic one. Every divisor of 360 is built by choosing how many 2s to include (0, 1, 2 or 3 — four choices), how many 3s (three choices) and how many 5s (two choices). Each combination gives a different divisor, and every divisor comes from exactly one combination, so the total is the product of the choice counts.

The "plus one" is where people slip, and it is the zero exponent they forget. A divisor is allowed to use none of a prime — that is how 1, 5, 9 and 45 all qualify as divisors of 360 despite missing the 2s entirely. Leave the plus one out and you are counting only divisors that use every prime at least once, which is a different and much smaller set.

This calculator takes three exponents because three distinct primes covers most numbers a reader will bring to it. If yours has only one or two, set the unused slots to 1 — but read that carefully: an exponent of 1 contributes a factor of 2, meaning a prime that IS present to the first power. To genuinely omit a prime you would want an exponent of 0, which is outside this form's range. For $1001 = 7 \cdot 11 \cdot 13$ all three exponents are 1 and the answer is $2 \times 2 \times 2 = 8$, which is right. For $12 = 2^2 \cdot 3$ you would work with the two-prime version by hand: $3 \times 2 = 6$.