2018 UNCO Math Contest II Problems/Problem 4
How many positive integer factors of are not perfect squares?
We can use complementary counting. Taking the prime factorization of , we get .So the total number of factors of is factors. Now we need to find the number of factors that are perfect squares. So back to the prime factorization, . Now we get factors that are perfect squares. So there are positive integer factors that are not perfect squares.
There is a similar way to the previous solution. The prime factorization of is . We need to find the amount of factors of that, so add to each exponent and multiply to get factors. Now we need to find the number of factors that are perfect squares. Perfect squares are numbers in the prime factorization with exponents of , etc. You find the max amount of the exponent that is less than the exponent in the prime factorization. There is a trick to that. You take the exponent in the prime factorization, for example 4. You divide by 2 and add 1 to the result to find the perfect squares in the number and exponent in the prime factorization. You also round up and do not add 1 if your answer is a decimal. You do to get 5, to get 2, and and round up to get . Multiply those answers to get 5^2^4 to get 40 perfect squares. Subtract it from to get . So there are positive integer factors that are not perfect squares.
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