Find all positive and negative factors, factor pairs, divisor sum, prime factorization, and number classification for any integer.
A factor (or divisor) of an integer n is any integer d that divides n without leaving a remainder (i.e. n mod d = 0). Factors naturally come in complementary pairs (d, n/d). Factoring integers reveals whether a number is prime, composite, square, or perfect.
Every divisor d has a partner n/d. Therefore, one only needs to test potential factors up to the square root √n; all paired factors above √n are automatically found.
A prime number has exactly two distinct positive divisors: 1 and itself (e.g. 2, 3, 5, 7, 11). A composite number has three or more divisors.
Only perfect squares have an odd number of factors (e.g. 36 has 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, 36) because the central factor 6 pairs with itself (6 × 6 = 36).
A number is "perfect" if the sum of its proper divisors (excluding the number itself) exactly equals the number (e.g. 6 = 1+2+3, 28 = 1+2+4+7+14).
List all positive factors of 28, find its factor pairs, and check if 28 is a perfect number.
The number 1 is neither prime nor composite. It is classified as a unit, because it has only 1 positive divisor.
Yes, in formal number theory, every positive factor has a negative counterpart (e.g. -1 × -28 = 28). In elementary math, "factors" usually refers specifically to positive factors.
100 = 2² × 5². Number of factors d(100) = (2+1)(2+1) = 3 × 3 = 9 factors: 1, 2, 4, 5, 10, 20, 25, 50, 100.
If the sum of proper divisors exceeds the number, it is abundant (e.g. 12: 1+2+3+4+6 = 16 > 12). If the sum is less than the number, it is deficient (e.g. 8: 1+2+4 = 7 < 8).
Understanding your calculation
A positive integer factor divides the entered integer exactly. Factor pairs multiply back to the original number.
The positive factors of 12 are 1, 2, 3, 4, 6 and 12; the pairs are 1 × 12, 2 × 6 and 3 × 4.
Factor listing and prime factorization are different outputs. Zero has infinitely many integer divisors, so it needs special handling.