Lisa,

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).

Penny

Lisa,

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).

Harley

Hi Lisa,

If the prime factorization of n is p₁^e₁ x p₂^e₂ x ... x p_k^e_k, then
tau(n) = (e₁ + 1)(e₂ + 1)...(e_k + 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.

Victoria

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 3^a × 7^b 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.

Harley