Case 1
- Number
- 360
24
Open with these valuesDivisor Count Calculator
24divisors
Result: 24 divisorsThe answer is τ(n): how many positive whole numbers divide n exactly, counting both 1 and n itself. Exactly two means n is prime. 360 has 24 divisors, which is why it carries the degrees of a circle. The sum of those divisors, σ(n), stands beside every row of the table.
The numbers at a glance
| Number | Result |
|---|---|
| 100 | 9 |
| 200 | 12 |
| 300 | 18 |
| 360Your value | 24 |
| 400 | 15 |
| 500 | 12 |
| 600 | 24 |
| 700 | 18 |
Worked examples
How it's calculated
τ(n) = how many whole numbers divide n exactly
- StepEnter a whole number from 1 to one billion.
- StepTest every i from 1 up to the square root of n.
- StepEach i that divides n exactly gives two divisors, i and n ÷ i.
- ResultA perfect square counts its root once — that is why 100 has an odd 9.
Reference table
| n | Sum σ(n) | Divisors τ(n) |
|---|---|---|
| 1 | 1 | 1 |
| 12 | 28 | 6 |
| 17 | 18 | 2 |
| 28 | 56 | 6 |
| 100 | 217 | 9 |
| 360 | 1170 | 24 |
| 1000000 | 2480437 | 49 |
Questions
What is a divisor of a number?
A divisor of a positive whole number n is any positive whole number that divides n with no remainder. The divisors of 12 are 1, 2, 3, 4, 6 and 12. Their count is written τ(n) and their sum σ(n).
How do I count the divisors of a number?
Test every whole number from 1 up to the square root of n; whenever i divides n exactly, both i and n ÷ i are divisors, giving you two at once. If i equals n ÷ i, count it only once. For 36 you test 1 to 6 and end up with 9 divisors.
How does the divisor count tell me a number is prime?
A prime has exactly two divisors, 1 and itself, so a count of 2 means prime. A count of 1 happens only for n = 1, which is neither prime nor composite. Anything above 2 is composite.
What is a perfect number?
A perfect number equals the sum of its divisors other than itself, so σ(n) = 2n. The smallest are 6 and 28; the table shows σ(28) = 56, which is exactly twice 28.
Why does 360 have so many divisors?
Because 360 = 2³ × 3² × 5, its divisor count is (3+1) × (2+1) × (1+1) = 24. Numbers built from many small primes carry the most divisors for their size, which is why 360 became the number of degrees in a circle.
Sources and last check
Information, not professional advice.