Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 4"
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+ | == Solution 2 == | ||
+ | There is a similar way to the previous solution. The prime factorization of <math>36,000,000 is </math>2^8\cdot3^2\cdot5^6<math>. We need to find the amount of factors of that, so add </math>1 to each exponent and multiply to get <math>(8+1)(2+1)(6+1) = 189</math> 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 <math>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 </math>8/2 + 1<math> to get 5, </math>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 <math>189 to get </math>149. So there are <math>\boxed{149}</math> positive integer factors that are not perfect squares. | ||
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+ | ~Aarushgoradia18 | ||
== See also == | == See also == |
Revision as of 19:52, 21 December 2021
Contents
Problem
How many positive integer factors of are not perfect squares?
Solution
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.
~Ultraman
Solution 2
There is a similar way to the previous solution. The prime factorization of 2^8\cdot3^2\cdot5^6
1 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
8/2 + 1
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
149. So there are
positive integer factors that are not perfect squares.
~Aarushgoradia18
See also
2018 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |