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Divisors Generator

List all positive divisors of N. Shows total count and whether N is prime (only 2 divisors) or perfect.

Number N (1 to 1,000,000,000)

Quantidade de divisores: — · Soma: — ·

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Every divisor, and what their sum reveals

The divisors of a number are the integers dividing it without remainder. Listing them by hand gets tedious quickly: 360 has 24, and 720 has 30. The page finds them all and also sums them — because that sum sorts the number into one of three categories the Greeks were already studying, and which still produce open problems.

The search only runs to the square root, and that is where the saving is. Divisors come in pairs multiplying to the number: if 4 divides 100, then 25 divides it too, because 4 times 25 is 100. Find the smaller of each pair and the larger comes free. The one special case is a perfect square, where the root pairs with itself and must not be counted twice.

Comparing the sum of proper divisors — all of them except the number itself — against the number gives the three categories. If the sum equals the number, it is perfect, and 6 is the first: 1 plus 2 plus 3. If it is more, the number is abundant, like 12. If less, deficient, which covers most numbers, including every prime, whose proper divisors always sum to 1.

Frequently asked questions

Why do divisors come in pairs?

Because division is symmetric: if a divides n, then n divided by a also divides n. The two multiply to n, and one of them is necessarily at or below the root. That is why walking up to the root finds them all.

Which number has the most divisors?

There is no maximum — you can always build numbers with more. The ones setting records for their size are called highly composite, and 12, 60, 360 and 2520 are among them. It is no coincidence that 60 and 360 show up in clocks and angles: many divisors means many exact divisions.

Do amicable numbers fit into this?

They do, as the two-number variation: a pair where the proper divisors of each sum to the other. The pair 220 and 284 is the smallest, known since antiquity. A perfect number, in that sense, is amicable with itself.

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