What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors?

The numbers will all be of the form p^4, where p is a prime.





2^4 = 16


3^4 = 81


5^4 = 625





Add them up:


16 + 81 + 625 = 722





Answer:


722





P.S. For anyone that is wondering why this is the case, the number of factors is related to the indices in the prime factorization. For example if I had the prime factorization of 75, that would be 3 x 5虏





In forming factors of 75, 3 would be included or not included (2 cases). And 5 would be included 0, 1 or 2 times (3 cases). 2 x 3 = 6, so 75 has 6 factors {1, 3, 5, 15, 25, 75}.





For something to have *five* factors, it must have an exponent of 4 on a single prime factor, and nothing else. That prime factor can then be used 0, 1, 2, 3 or 4 times (5 cases). So all numbers that have exactly 5 factors will be of the form p^4.





In general, you can represent the prime factorization as:


p1^i x p2^j x p3^k x ...





The exponents are i, j, k, etc. Add one to each of these and multiply and you'll have the number of factors. For example 600 = 2 x 2 x 2 x 3 x 5 x 5 = 2^3 x 3^1 x 5^2. The number of factors will be 4 x 2 x 3 = 24.What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors?
First we need the numbers less than 1000, with exactly five positive integer divisors. Does 1 count as an integer divisor? Does ';exactly'; also mean unique?





Hmmm ... looks like there are no such numbers.


If the number has two unique prime factors (A and B), then it has the divisors:


1, A, B, AB = 2, 3, or 4 divisors, depending on how you count them.





If the number has three unique prime factors, then it has the divisors:


1, A, B, C, AB, AC, BC, ABC = 6, 7, or 8 divisors depending on how you count them.





So lets check out numbers with non-unique prime factors.


If the number is AAB, then the factors are:


1, A, B, AA, AB, and AAB. That's 4, 5, or 6.





Hmmm ... %26lt;more calculations%26gt;


How about these:


16: factors are 1, 2, 4, 8, and 16


81: factors are 1, 3, 9, 27, and 81


625: factors are 1, 5, 25, 125, and 625





Sum is 16 + 81 + 625 = 722What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors?
I wrote a computer program to figure this out. The three numbers are


16, 81, and 625. What they all have in common (as far as I can tell) is that they are all fourth powers of a prime number. Anyway, the sum of these three numbers is 722.