Be Quiet Children!Math brain teasers require computations to solve.
One day, a frustrated math teacher lost his patience with his students' non-stop chatting. Thus, he decided to give the ultimate hard problem:
Find the only positive integer number less than 20,000 that is also a sum of three positive integers, all containing exactly 7 factors.
There was a steady silence. Can you break the silence by figuring it out?
HintFor the prime factorization of positive integer x, we get
x=[y(sub 1)^K (sub 1)][y(sub 2)^ K (sub 2)]..., where y sub(n) is a prime number, and K sub(n) is the power to which y sub(n) is raised, we can figure out the number of prime factors by:
[K(sub 1) + 1][K(sub 2) + 1]...
For example, the prime factorization of 228 is (2^2)(3^1)(19^1). 228 has 12 factors, since (2+1)(1+1)(1+1) = 12.
First, you need to figure out the separate integers. If you have read the hint, you would know how to use prime factorization to your advantage. The only way to get 7 as the number of factors is 1 x 7. So the prime factorization of any number with 7 factors is n^6 times x^0, where n and x are both primes. But if you look, x^0, for all nonzero values of x, is 1! So, any number with exactly 7 factors must be a prime raised to the sixth power. This yields 2^6 +3^6+5^6 = 16418. Any prime 7 or greater to the sixth power is greater than 20,000.
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