One Methods:Why This Works
1, 2, 3, 4, 6, 8, 12, 24 are all divisors (or factors) of 24. You can divide 24 by any of them and you will arrive at a whole number. So there are 8 divisors of the number 24. This article tells you how to calculate this number quickly.
1As an example, we calculate the number of divisors of 24. First we factor the number. 24 = 23*3Ad
2Note the exponents. These are 3 and 1. When a number doesn’t have an exponent its exponent equals 1.
3Add one to each of the exponents and multiply. (3+1)*(1+1) = 4*2 = 8. The number of divisors is 8.
Why This Works
- This is not a rigorous proof, it’s an explanation.
- A number d is a divisor of 24 when d is a positive integer, and the result of the division, n, is an integer. This means that any d must be of the form 2p*3q, where p and q are integers. Furthermore p<=3, q<=1, otherwise n would still not be an integer. p and q are nonnegative, otherwise d is not an integer.
- Remember that any nonzero number raised to the zeroth power equals 1. So p and q can also be 0. So p can be 0, 1, 2, or 3 which gives for 4 possibilities for p. q can be 0 and 1, which gives 2 possibilities for q. The number of possibilities is just one larger than the corresponding exponent.
- Because these possibilities are independent we can multiply them in order to arrive at the total number of divisors.
- 17 = 171 --> 2
- 25 = 52 --> 3
- 60 = 22*3*5 --> 12
- 100 = 22*52 --> 9
- 45360 = 24*34*5*7 --> 100
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- When the number is a square, the number of divisors will be odd. When it's not a square then the number of divisors will be even.
In other languages:
Español: Cómo determinar el número de divisores de un entero, Русский: найти количество делителей целого числа, Italiano: Come Determinare il Numero di Divisori di un Numero Intero, Português: Como Determinar a Quantidade de Divisores de um Número Inteiro
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