### Brain Teasers

# Be Quiet Children!

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?

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?

### Hint

For the prime factorization of positive integer x, we getx=[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.

### Answer

16,418.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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