2016 AMC 8 Problems/Problem 15
Contents
Problem
What is the largest power of that is a divisor of ?
Solution 1
First, we use difference of squares on to get . Using difference of squares again and simplifying, we get . Realizing that we don't need the right-hand side because it doesn't contain any factor of 2, we see that the greatest power of that is a divisor is .
~CHECKMATE2021
Solution 2 (a variant of Solution 1)
Just like in the above solution, we use the difference-of-squares factorization, but only once to get We can then compute that this is equal to Note that (we don't need to factorize any further as is already odd) thus the largest power of that divides is only while so the largest power of that divides is Hence, the largest power of that is a divisor of is
Solution 3 (Lifting the exponent)
Let We wish to find the largest power of that divides .
Denote as the largest exponent of in the prime factorization of . In this problem, we have .
By the Lifting the Exponent Lemma on ,
Therefore, exponent of the largest power of that divids is so the largest power of that divides this number is .
~CHECKMATE2021
Solution 4 (Brute Force)
We can simply take 13 to the 4th power, which is 28561. We subtract that by 11 to the 4th power, which is 14641 (You can use Pascal's Triangle to find this). Finally, subtract the numbers to get 13920.
To test the options, since we need the largest one, we can go from top down. Testing, we see that both D and E are decimals, and 32 works. So, our answer is
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Note: This is not the fastest way, and is not recommended.
I completely agree even though I did it./reader
Video Solution by OmegaLearn
https://youtu.be/HISL2-N5NVg?t=3705
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Video Solution
~savannahsolver
Video Solution (CREATIVE THINKING!!!)
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See Also
2016 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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All AJHSME/AMC 8 Problems and Solutions |
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