Find All Divisors

Divisors come in pairs: if d divides n, then so does n/d, and exactly one member of each pair is ≤ √n. So instead of looping from 1 to n (which would take a billion iterations for the upper limit), the tool loops from 1 to floor(√n) – about 31,623 iterations at the max – and pushes both i and n/i when i divides n. When i² = n (perfect square) we skip the duplicate.

A positive integer greater than 1 whose only divisors are 1 and itself. So the divisor count is exactly 2. The tool detects this and shows a “prime” badge. (1 is not prime by convention – it has only 1 divisor.)

A positive integer that equals the sum of its proper divisors (divisors excluding itself). The first four are 6 (= 1+2+3), 28 (= 1+2+4+7+14), 496, and 8128. The tool detects perfect numbers and shows a “perfect number” badge. Perfect numbers below 1 billion are vanishingly rare – there are only five of them.

Compare the sum of proper divisors (s) to the number n. If s < n, the number is deficient (most numbers are). If s = n, it’s perfect. If s > n, it’s abundant. 12 is the smallest abundant number: 1+2+3+4+6 = 16 > 12. The badge shows which class your number falls into.

It’s a UI-time bound, not a math limit. √(10⁹) ≈ 31,623 – a single tight loop runs in well under 10 ms in any modern JS engine. √(10¹⁸) would be 10⁹ iterations – about 5-10 seconds, which would freeze the page. If you need bigger numbers, run the same algorithm in a Web Worker or on the command line.

For each divisor pair (a, b) where a ≤ b and a × b = n, one line. For 12: 1 × 12, 2 × 6, 3 × 4. Useful for factoring intuition and for showing rectangular tilings (a 12-cell grid can be 1×12, 2×6, or 3×4).

For typical numbers near a billion, the divisor count is small (most have 4-16 divisors). The “highly composite” numbers – those with more divisors than any smaller number – have the most. Examples: 60 has 12 divisors, 5040 has 60, 720720 has 240. Near 1 billion, the highly composite champion is 735134400 with 1344 divisors.

No. The page is three static files (HTML, CSS, JS) and runs entirely in your browser using plain arithmetic. No fetch, no XHR, no analytics, no cookies. You can disconnect from the internet after the page loads and the tool still works.