



 
Lisa, There are infinitely many such numbers like 3^{2} x 7^{16} or 7^{2} x 29^{16} (any number of the form p^{2} x q^{16} 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. 2^{50}, 3^{50}, 5^{50} (any number of the form p^{50} where p is a prime). Harley Hi Lisa, If the prime factorization of n is p_{1}_{}^{e1 }x p_{2}_{}^{e2} x ... x p_{k}_{}^{ek}, then For example, suppose you wanted to find n so that tau(n) = 35. Two different solutions are 2^{34} and 2^{4} x 3^{6}. There are many different solutions that can be obtained by replacing the 2 and / or 3 by other prime numbers. Victoria Lisa wrote back
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 11000, 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 3^{2} x 7^{16} 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 7^{16} but it does give 7^{8} = 5764801 so 3^{2} x 7^{16} = (3 × 5764801)^{2} = 9 × 5764801 × 5764801. Why does this work? The only prime divisors of 3^{2} x 7^{16} are 3 and 7 and hence any divisor of 3^{2} x 7^{16} must also have 3 and 7 as its only prime divisors. (To make this work nicely I am going to use the convention that 3^{0} = 7^{0} = 1.) Hence a divisor of 3^{2} x 7^{16} is of the form 3^{a} × 7^{b} where a and b are some nonnegative 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 3^{2} x 7^{16}. Harley  


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