How this calculation works
Every positive divisor chooses an exponent from zero through e for each prime. Multiply the numbers of choices to count divisors; multiply the small geometric sums to add them.
n = ∏p^e; divisor count = ∏(e+1); divisor sum = ∏(p^(e+1)−1)/(p−1)A worked example
360 = 2^3 × 3^2 × 5. It has 4 × 3 × 2 = 24 positive divisors, whose sum is 1,170.
Good to know
Accepts integers 2 through one billion. This is exact trial division, not a large-number factoring service. The sum includes 1 and n; no negative divisors are counted.
Reference: OpenStax: the fundamental theorem of arithmetic (opens in a new tab) ↗
One factorization answers several questions
The fundamental theorem of arithmetic says every integer above one has a unique prime-power factorization, apart from factor order. For 360, repeated exact division finds 2^3 × 3^2 × 5. The entries in the table are those prime powers, not a list of all positive divisors.
A divisor of 360 may use 0, 1, 2 or 3 copies of 2; 0, 1 or 2 copies of 3; and 0 or 1 copy of 5. Multiplying those independent choices gives 4 × 3 × 2 = 24 divisors.
The Euler totient calculator uses the same distinct prime factors but counts coprime integers instead of positive divisors.
Counting is different from adding
To add every divisor, add each prime’s possible powers first. The three sums are 1+2+4+8 = 15, 1+3+9 = 13, and 1+5 = 6. Their product is 1,170. This is why the divisor sum cannot be obtained by multiplying the divisor count by the original number.
The Euler totient calculator instead counts numbers coprime to the input. For 360, that asks a different question from counting factors of 360. A gcd calculator checks whether one particular pair shares a factor.
Keep zero and negative divisors outside the convention
This page counts positive divisors, so 1 and the number itself are included. Zero is never a divisor of a nonzero integer. Negative divisors are excluded deliberately, rather than silently doubling the displayed count.
Factorization by trial division is practical for the declared one-billion limit. It is not a general algorithm for factoring enormous cryptographic integers; the visible bound makes that difference clear.
A few questions, answered
Is one prime?
No. One has no prime factorization and is outside this tool’s input range.
Why does a repeated prime appear once in the table?
Its exponent records how many times that prime divides the number.
Does the divisor sum include the number itself?
Yes, as well as 1 and every other positive divisor.