Factor Calculator
The factor calculator lists every factor of a number in the pairs that multiply to it, with the prime factorization, the divisor count, and whether it is prime, perfect or abundant.
Factors
- How many From the exponents: (4 + 1) x (2 + 1) = 15.
- 15
- Prime factorization
- 2^4 x 3^2
- Prime factors only
- 2, 3
- Composite factors
- 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72
- Sum of the proper divisors The proper divisors add to more than 144, which makes it abundant.
- 259 (abundant)
| Smaller | Larger | Product |
|---|---|---|
| 1 | 144 | 1 x 144 = 144 |
| 2 | 72 | 2 x 72 = 144 |
| 3 | 48 | 3 x 48 = 144 |
| 4 | 36 | 4 x 36 = 144 |
| 6 | 24 | 6 x 24 = 144 |
| 8 | 18 | 8 x 18 = 144 |
| 9 | 16 | 9 x 16 = 144 |
| 12 | 12 | 12 x 12 = 144 |
How the Factor Calculator works
A factor of a number divides it with no remainder. The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72 and 144, fifteen in all.
The efficient way to find them all is to notice that factors come in pairs. If 8 divides 144 then so does 18, because 8 x 18 = 144. Every factor below the square root has exactly one partner above it, so testing divisors from 1 up to sqrt(144) = 12 finds all of them in twelve steps rather than 144. For a nine-digit number that is the difference between thirty thousand tests and a billion.
That pairing means only a perfect square has an odd number of factors, since its square root pairs with itself, so you can tell whether a number is a perfect square by counting its factors, without taking a square root.
The prime factorization organizes all of this. 144 = 2^4 x 3^2, and every factor of 144 is 2 to some power from 0 to 4, times 3 to some power from 0 to 2. That gives 5 x 3 = 15 combinations, which is the factor count, arrived at by multiplying one more than each exponent. The prime factorization is therefore a complete generator of the factor list and not merely a related fact.
A "common factor" of two numbers divides both, and the largest is the GCF; a "common multiple" is divisible by both, and the smallest is the LCM.
Three classifications come from adding up a number's proper divisors (all its factors except itself), and they are the oldest questions in number theory. If the sum equals the number, it is perfect, as 6 = 1 + 2 + 3 is. If the sum exceeds the number, it is abundant: 12's proper divisors are 1, 2, 3, 4 and 6, adding to 16. If the sum falls short, it is deficient, and every prime is as deficient as possible since its only proper divisor is 1.
Two practical uses. Factoring tells you the ways to arrange a quantity in a rectangle: 144 objects fit in a 1x144, 2x72, 3x48, 4x36, 6x24, 8x18, 9x16 or 12x12 grid, which is the factor-pair list. And in algebra, factoring a quadratic by inspection means finding the factor pair of the constant term that adds to the middle coefficient. For x^2 + 25x + 144 that is the pair 9 and 16, giving (x + 9)(x + 16).
Factor Calculator FAQ
What are all the factors of 144?
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72 and 144: fifteen in total. They pair up to give the product: 1x144, 2x72, 3x48, 4x36, 6x24, 8x18, 9x16, and 12x12. The 12 pairs with itself, which is why the count is odd.
Why do perfect squares have an odd number of factors?
Because the square root pairs with itself instead of with a different number. For 144, the pair 12x12 contributes one factor rather than two, so the total is odd. Every number that is not a perfect square has its factors in distinct pairs and therefore an even count.
What is the difference between a factor and a multiple?
Factors divide into the number; multiples are divisible by it. 12 is a factor of 144 and 144 is a multiple of 12, the same relationship from each end. A number has finitely many factors, all no larger than itself, and infinitely many multiples.
What is a perfect number?
One whose proper divisors (every factor except itself) add up to the number. 6 is perfect because 1 + 2 + 3 = 6, and so are 28, 496 and 8,128. Only 51 are known, all even, and whether an odd one exists has been open since antiquity.