2018 UNCO Math Contest II Problems/Problem 4

Problem

How many positive integer factors of $36,000,000$ are not perfect squares?

Solution

We can use complementary counting. Taking the prime factorization of $36,000,000$ , we get $2^8\cdot3^2\cdot5^6$ .So the total number of factors of $36,000,000$ is $(8+1)(2+1)(6+1) = 189$ factors. Now we need to find the number of factors that are perfect squares. So back to the prime factorization, $2^8\cdot3^2\cdot5^6 = 4^4\cdot9^1\cdot5^3$ . Now we get $(4+1)(1+1)(3+1)=40$ factors that are perfect squares. So there are $189-40=\boxed{149}$ positive integer factors that are not perfect squares.

~Ultraman

Solution 2

There is a similar way to the previous solution. The prime factorization of 36,000,000 $is$ 2^8\cdot3^2\cdot5^6 $. We need to find the amount of factors of that, so add$ 1 to each exponent and multiply to get $(8+1)(2+1)(6+1) = 189$ 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 $0, 2, 4, 6, etc. You find the max amount of the exponent that is less than the exponent in the prime factorization. There is a trick to that. You take the exponent in the prime factorization, for example 4. You divide by 2 and add 1 to the result to find the perfect squares in the number and exponent in the prime factorization. You also round up and do not add 1 if your answer is a decimal. You do$ 8/2 + 1 $to get 5,$ 2/2 + 1 to get 2, and 7/2 and round up to get 4. Multiply those answers to get 5^2^4 to get 40 perfect squares. Subtract it from $189 to get$ 149. So there are $\boxed{149}$ positive integer factors that are not perfect squares.

~Aarushgoradia18

2018 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2018 UNCO Math Contest II Problems/Problem 4

Contents

Problem

Solution

Solution 2

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