Math CentralQuandaries & Queries


Question from Lisa, a teacher:

What number has a Tau which equals 51? In other words, what number has exactly 51 divisors? It must be a square of some kind!


There are infinitely many such numbers like 32 x 716 or 72 x 2916 (any number of the form p2 x q16 where p&q are primes). The key is that 51 = 3 × 17 = (2 + 1) × (16 + 1).



In case Penny's infinitely many aren't enough here are infinitely many more. 250, 350, 550 (any number of the form p50 where p is a prime).


Hi Lisa,

If the prime factorization of n is p1e1 x p2e2 x ... x pkek, then
tau(n) = (e1 + 1)(e2 + 1)...(ek + 1). You need to choose some different primes and some exponents so that product in the previous sentence equals 51.

For example, suppose you wanted to find n so that tau(n) = 35. Two different solutions are 234 and 24 x 36. There are many different solutions that can be obtained by replacing the 2 and / or 3 by other prime numbers.


Lisa wrote back

Question from Lisa, a teacher:

I asked what number has exactly 51 divisors, but what I left out was that it can not be a prime raised to a power. That was my first choice as well, but we are limited. Now what? Her answer includes a prime (2) raised to a power (50). I can't have that. I'm trying to find the actual number. I was told it was a hit and miss problem where there is no formula to it. For example, I tried 3600 and 8100 both of which have 45 divisors. I found a website that gave the number of factors 1-1000, but no number had 51 factors.

Hi again Lisa,

There is a theory for this, it is what is in Victoria's response. Penny's response 32 x 716 will meet your requirement of not being a prime raises to a power. If you want the number itself not written as powers of primes then you can calculate it. My calculator won't give the exact value for 716 but it does give 78 = 5764801 so 32 x 716 = (3 × 5764801)2 = 9 × 5764801 × 5764801.

Why does this work? The only prime divisors of 32 x 716 are 3 and 7 and hence any divisor of 32 x 716 must also have 3 and 7 as its only prime divisors. (To make this work nicely I am going to use the convention that 30 = 70 = 1.) Hence a divisor of 32 x 716 is of the form 3a × 7b where a and b are some non-negative integers. In fact I know more, a can only be 0, 1, or 2 and b can only be 0, 1, 2, ..., 16. Hence there are 3 choices for a and 17 choices for b. Hence there are 3 × 17 = 51 divisors of 32 x 716.


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