What is 23 divisible by?
23 has 2 divisors:
1, 23
Arithmetic properties of 23
τ(23)
=
2
Number of divisors
σ(23)
=
24
Sum of divisors
23 is prime — its only divisors are 1 and itself.
23 is not a perfect square.
Find divisors of another number
Integers from 1 to 10¹² (1,000,000,000,000).
About divisors
What is a divisor?
A divisor (or factor) of a positive integer n is any positive integer d such that n ÷ d leaves no remainder (n mod d = 0). For example, the divisors of 12 are 1, 2, 3, 4, 6, 12 — there are 6 of them.
How is the number of divisors counted?
The divisor function τ(n) (tau) counts all positive divisors of n. If n = p₁^e₁ × p₂^e₂ × … × pₖ^eₖ, then τ(n) = (e₁+1)(e₂+1)…(eₖ+1). For 360 = 2³ × 3² × 5¹: τ(360) = 4 × 3 × 2 = 24.
What is the sum of divisors σ(n)?
The sum-of-divisors function σ(n) (sigma) is the total of all positive divisors of n, computed as σ(n) = ∏ (1 + p + p² + … + p^e) for each prime-power factor. For 360 = 2³ × 3² × 5: σ(360) = 15 × 13 × 6 = 1,170.
Frequently asked questions
Loop from 1 to √n. For each i that divides n exactly, both i and n/i are divisors. This takes at most √(10¹²) = 10⁶ steps, fast enough for any value in this range.
A perfect number is a positive integer equal to the sum of its proper divisors. The smallest is 6 = 1 + 2 + 3. The next are 28, 496, and 8,128. Only 51 perfect numbers are currently known.
An abundant number is one where the sum of its proper divisors exceeds the number itself. For example, 12 has proper divisors 1, 2, 3, 4, 6 whose sum 16 > 12, so 12 is abundant.
Divisors come in pairs (d, n/d). When n is a perfect square, the divisor at the exact square root is paired with itself and counted once — giving an odd total. For example, 36 has 9 divisors (odd).
The common divisors of two integers a and b are exactly the divisors of GCD(a, b). Also, LCM(a, b) = a × b / GCD(a, b).