Difference between revisions of "2011 AMC 8 Problems/Problem 17"
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+ | ==Problem== | ||
Let <math>w</math>, <math>x</math>, <math>y</math>, and <math>z</math> be whole numbers. If <math>2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588</math>, then what does <math>2w + 3x + 5y + 7z</math> equal? | Let <math>w</math>, <math>x</math>, <math>y</math>, and <math>z</math> be whole numbers. If <math>2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588</math>, then what does <math>2w + 3x + 5y + 7z</math> equal? | ||
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==Solution== | ==Solution== | ||
+ | The [[prime factorization]] of <math>588</math> is <math>2^2\cdot3\cdot7^2.</math> We can see <math>w=2, x=1,</math> and <math>z=2.</math> Because <math>5^0=1, y=0.</math> | ||
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+ | <cmath>2w+3x+5y+7z=4+3+0+14=\boxed{\textbf{(A)}\ 21}</cmath> | ||
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+ | ==Video Solution== | ||
+ | https://youtu.be/PxBKpg-HKu8 ~David | ||
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+ | ==Video Solution 2== | ||
+ | https://youtu.be/5vpKkAue8Is. Soo, DRMS, NM | ||
+ | |||
+ | ==Video Solution 3== | ||
+ | https://youtu.be/_4KL96b9vbY | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution 4 by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=mYn6tNxrWBU | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2011|num-b=16|num-a=18}} | {{AMC8 box|year=2011|num-b=16|num-a=18}} | ||
+ | {{MAA Notice}} |
Latest revision as of 14:08, 17 December 2023
Contents
Problem
Let , , , and be whole numbers. If , then what does equal?
Solution
The prime factorization of is We can see and Because
Video Solution
https://youtu.be/PxBKpg-HKu8 ~David
Video Solution 2
https://youtu.be/5vpKkAue8Is. Soo, DRMS, NM
Video Solution 3
~savannahsolver
Video Solution 4 by SpreadTheMathLove
https://www.youtube.com/watch?v=mYn6tNxrWBU
See Also
2011 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.