A worked example: the factors of 24
Run 24 through the Factor Calculator and it returns 1, 2, 3, 4, 6, 8, 12, 24 — eight factors in total, and it is marked as not prime. Notice how the list is symmetric around the square root of 24 (about 4.9): 1 pairs with 24, 2 pairs with 12, 3 pairs with 8, and 4 pairs with 6. Every factor smaller than the square root has a partner larger than it, which is exactly why the list always comes out in matching pairs for a non-square number.
The formula: checking divisor pairs up to the square root
The calculator does not test every number from 1 to n. It only checks candidate divisors i up to the square root of n, and whenever i divides n evenly it records both i and its partner n divided by i. Feed it 100 and it stops checking once i passes 10 (the square root of 100), yet still returns the complete list: 1, 2, 4, 5, 10, 20, 25, 50, 100 — nine factors. Notice 100 has an odd number of factors, unlike 24's even count of eight. That happens because 10 pairs with itself (10 × 10 = 100): a perfect square always contributes one unpaired middle factor, which is why its factor count comes out odd instead of even.
The special case of a prime number: 17
Run 17 through the same calculator and the list shrinks to just 1, 17 — two factors, and it is flagged as prime. This is the defining property of a prime number: it has no divisor pair besides 1 and itself, because no whole number between 2 and 16 divides 17 evenly. Any number with more than two factors, like 24 or 100 above, is automatically composite.
Why the factor count matters
Whether a number's factor count comes out even or odd is a free signal about the number itself. 24 (eight factors) and 17 (two factors) both have even counts because their divisors pair up completely with no leftover. 100 is the odd one out here with nine factors, and that oddness is not a coincidence — it is a direct consequence of 100 being a perfect square (10 × 10). In fact, a whole number has an odd number of factors if and only if it is a perfect square; every other whole number will always show an even count, because every divisor pair has two distinct members.