Factors of 360, worked by hand
A factor of a number divides it with nothing left over. 360 ÷ 8 = 45 exactly, so 8 is a factor of 360. 360 ÷ 7 leaves 3, so 7 is not.
The fastest hand method is the division ladder. Write the number, divide by the smallest prime that goes in, write the answer underneath, and repeat until you reach 1. Load sample puts 360 in the box and draws this ladder:
Read the left column top to bottom: 2, 2, 2, 3, 3, 5. That gives 360 = 23 × 32 × 5. Every factor of 360 is some mix of those primes, such as 2 × 3 × 5 = 30 or 23 × 32 = 72.
Count the factors before you list them
Add 1 to each exponent in the prime factorization, then multiply. For 360 the exponents are 3, 2 and 1, so the count is (3 + 1)(2 + 1)(1 + 1) = 24. Each factor picks 0 to 3 twos, 0 to 2 threes and 0 or 1 five, and those choices multiply out.
This tells you when a hand-written list is finished. Stop searching once you hit the count.
| Number | Prime factorization | Exponent rule | Factors |
|---|---|---|---|
| 12 | 22 × 3 | 3 × 2 | 6 |
| 36 | 22 × 32 | 3 × 3 | 9 |
| 60 | 22 × 3 × 5 | 3 × 2 × 2 | 12 |
| 100 | 22 × 52 | 3 × 3 | 9 |
| 360 | 23 × 32 × 5 | 4 × 3 × 2 | 24 |
| 5040 | 24 × 32 × 5 × 7 | 5 × 3 × 2 × 2 | 60 |
Note 60 against 100. The bigger number has fewer factors, because 60 spreads its exponents across more primes. Calendars and clocks use 12, 60 and 360 for this reason: they split into many equal parts.
Factors, prime factors and multiples
Homework mixes these three up more than anything else on this topic. Using 12:
- Factors go into 12: 1, 2, 3, 4, 6, 12. A finite list.
- Prime factors are the factors which are prime: 2 and 3. Written with repeats, 12 = 2 × 2 × 3.
- Multiples are what 12 goes into: 12, 24, 36, 48 and on forever.
"Divisor" means the same as factor in this setting. Some textbooks reserve "proper factor" or "proper divisor" for every factor except the number itself, and the calculator above reports that sum as its own figure.
Why an odd factor count means a perfect square
Factors arrive in pairs. For 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, and 6 × 6. The last pair repeats one number, so it adds a single factor to the list, not two. 36 ends up with 9 factors.
Only perfect squares have a pair like 6 × 6, so only perfect squares have an odd factor count. Check it on 144 with the Try chip: 15 factors, square root 12. This also explains why the pairs table stops at the square root. Past that point, every pair repeats one already written in reverse.
Perfect, abundant and deficient numbers
Add up a number's proper factors and compare the total with the number:
- Perfect: the total equals the number. 6 = 1 + 2 + 3. 28, 496 and 8128 come next, then 33,550,336.
- Abundant: the total runs over. 12 gives 1 + 2 + 3 + 4 + 6 = 16.
- Deficient: the total falls short. 8 gives 1 + 2 + 4 = 7. Every prime is deficient, since its only proper factor is 1.
The result card labels each number with one of these. Try 496 to see a perfect number break down as 24 × 31.
Where a factor list gets used
Simplifying a fraction
For 84/126, factor both: 84 = 22 × 3 × 7 and 126 = 2 × 32 × 7. The shared part is 2 × 3 × 7 = 42. Divide top and bottom by 42 to get 2/3.
Equal groups and rows
Twenty-four students into equal teams: the pairs of 24 are 1 × 24, 2 × 12, 3 × 8 and 4 × 6. Those are your only even splits. Twenty-three students have one option, because 23 is prime.
Grids and screen layouts
A 1920-pixel row splits evenly into any factor of 1920. 1920 = 27 × 3 × 5, giving 32 column widths with no leftover pixels.
What this calculator won't do
- The ceiling is 9,007,199,254,740,991 (253 − 1), the largest whole number JavaScript stores exactly. Past it, digits get rounded and the factors would be wrong, so the page refuses.
- Big primes take longest. The search is trial division up to the square root. A 16-digit prime, or a product of two 8-digit primes, takes up to about a second on a phone. Small or smooth numbers finish at once.
- No RSA-sized numbers. Keys with hundreds of digits need algorithms such as the quadratic sieve, not this page.
- Whole numbers only. Decimals and fractions have no factor list in this sense. 0 is out too, since every number divides 0.
- Long lists get trimmed on screen. 720720 has 240 factors, and some 16-digit numbers have tens of thousands. The first 400 show, a button reveals the rest, and Copy always takes the full list.
- A minus sign gets dropped. The factors of −12 are the factors of 12 with both signs. Tick "Include negative factors" to see them.