Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 4"

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== Solution ==
 
== Solution ==
We can use [[complementary counting]]. Taking the prime factorization of <math>36,000,000</math>, we get <math>2^8\cdot3^2\cdot5^6</math>.So the total number of factors of <math>36,000,000</math> is <math>(8+1)(2+1)(6+1) = 189</math> factors.  
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We can use [[complementary counting]]. Taking the prime factorization of <math>36,000,000</math>, we get <math>2^8\cdot3^2\cdot5^6</math>.So the total number of factors of <math>36,000,000</math> is <math>(8+1)(2+1)(6+1) = 189</math> factors. Now we need to find the number of factors that are perfect squares. So back to the prime factorization, <math>2^8\cdot3^2\cdot5^6 = 4^4\cdot9^1\cdot5^3</math>. Now we get <math>(4+1)(1+1)(3+1)=40</math> factors that are perfect squares. So there are <math>189-40=\boxed{149}</math> positive integer factors that are not perfect squares.
 
 
Now we need to find the number of factors that are perfect squares. So back to the prime factorization, <math>2^8\cdot3^2\cdot5^6 = 4^4\cdot9^1\cdot5^3</math>. Now we get <math>(4+1)(1+1)(3+1)=40</math> factors that are perfect squares.
 
 
 
So there are <math>189-40=\boxed149</math> positive integer factors that are not perfect squares.
 
  
 
~Ultraman
 
~Ultraman

Latest revision as of 14:02, 2 January 2020

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

See also

2018 UNCO Math Contest II (ProblemsAnswer KeyResources)
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
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