Difference between revisions of "2009 AMC 8 Problems/Problem 17"

Latest revision as of 06:08, 28 December 2023

Problem

The positive integers $x$ and $y$ are the two smallest positive integers for which the product of $360$ and $x$ is a square and the product of $360$ and $y$ is a cube. What is the sum of $x$ and $y$ ?

$\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 85 \qquad \textbf{(C)}\ 115 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 610$

Video Solution by OmegaLearn

https://youtu.be/7an5wU9Q5hk?t=2768

Video Solution (XXX)

https://www.youtube.com/watch?v=ZuSJdf1zWYw ~David

Solution

The prime factorization of $360=2^3 \cdot 3^2 \cdot 5$ . If a number is a perfect square, all of the exponents in its prime factorization must be even. Thus we need to multiply by a 2 and a 5, for a product of 10, which is the minimum possible value of x. Similarly, y can be found by making all the exponents divisible by 3, so the minimum possible value of $y$ is $3 \cdot 5^2=75$ . Thus, our answer is $x+y=10+75=\boxed{\textbf{(B)}\ 85}$ .

Solution 2 (Using Answer Choices)

From the question's requirements, we can figure out $x$ is $10$ . Then we can use the answer choices to find what $y$ is. Let's start with $A$ . If $A$ was right, then $y=70$ . We can multiply $70$ by $360$ and get $25200$ , which isn't a perfect cube. Then we move to $B$ . $85-10=75$ , so $y=75$ if $B$ is right. Then we multiply $75$ by $360$ to get $27000$ , which is $30^3$ . Therefore, our answer is $\boxed{\textbf{(B)}\ 85}$ because $y=75$ and $75+10=85$ .

~Trex226

2009 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 8: / Line 8: @@
 \textbf{(D)}\    165   \qquad
 \textbf{(E)}\    610</math>
+==Video Solution by OmegaLearn==
+https://youtu.be/7an5wU9Q5hk?t=2768
+==Video Solution (XXX)==
+https://www.youtube.com/watch?v=ZuSJdf1zWYw   ~David
 ==Solution==
-The prime factorization of <math>360=2^3*3^2*5</math>. If a number is a perfect square, all of the exponents in its prime factorization must be even. Thus we need to multiply by a 2 and a 5, for a product of 10, which is x. Similarly, y can be found by making all the exponents divisible by 3, so <math>y=3*5^2=75</math>. Thus <math>x+y=\boxed{\textbf{(B)}\ 85}</math>.
+The prime factorization of <math>360=2^3 \cdot 3^2 \cdot 5</math>. If a number is a perfect square, all of the exponents in its prime factorization must be even. Thus we need to multiply by a 2 and a 5, for a product of 10, which is the minimum possible value of x. Similarly, y can be found by making all the exponents divisible by 3, so the minimum possible value of <math>y</math> is <math>3 \cdot 5^2=75</math>. Thus, our answer is <math>x+y=10+75=\boxed{\textbf{(B)}\ 85}</math>.
+==Solution 2 (Using Answer Choices)==
+From the question's requirements, we can figure out <math>x</math> is <math>10</math>. Then we can use the answer choices to find what <math>y</math> is.
+Let's start with <math>A</math>. If <math>A</math> was right, then <math>y=70</math>. We can multiply <math>70</math> by <math>360</math> and get <math>25200</math>, which isn't a perfect cube. Then we move to <math>B</math>. <math>85-10=75</math>, so <math>y=75</math> if <math>B</math> is right. Then we multiply <math>75</math> by <math>360</math> to get <math>27000</math>, which is <math>30^3</math>. Therefore, our answer is <math>\boxed{\textbf{(B)}\ 85}</math> because <math>y=75</math> and <math>75+10=85</math>.
+~Trex226
 ==See Also==
 {{AMC8 box|year=2009|num-b=16|num-a=18}}
 {{MAA Notice}}

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