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Frequently Asked Questions

How many divisors does a number have?


Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs.

48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8

Hence we can see that 48 has exactly ten divisors. It should also be clear that, using this method, we only ever need to work from 1 up to the square root of the number.

Although this method is quick and easy with small numbers, it is tedious and impractical for larger numbers. Fortunately there is a quick and accurate method using the divisor, or Tau, function.

Let d(n) be the number of divisors for the natural number, n.

We begin by writing the number as a product of prime factors: n = paqbrc...
then the number of divisors, d(n) = (a+1)(b+1)(c+1)...

To prove this, we first consider numbers of the form, n = pa. The divisors are 1, p, p2, ..., pa; that is, d(pa)=a+1.

Now consider n = paqb. The divisors would be:

1pp2...pa
qpqp2q...paq
q2pq2p2q2...paq2
...............
qbpqbp2qb...paqb

Hence we prove that the function, d(n), is multiplicative, and in this particular case, d(paqb)=(a+1)(b+1). It should be clear how this can be extended for any natural number which is written as a product of prime factors.

The number of divisor function can be quickly demonstrated with the example we considered earlier: 48 = 24×31, therefore d(48)=5×2=10.