Difference between revisions of "2022 AMC 12B Problems/Problem 15"
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Revision as of 01:26, 18 November 2022
Problem: One of the following numbers is not divisible by any prime number less than 10. Which is it?
Solution 1 (Process of Elimination)
We examine option E first. has a units digit of (Taking the units digit of the first few powers of two gives a pattern of ) and has a units digit of (Taking the units digit of the first few powers of three gives a pattern of ). Adding and together, we get , which is a multiple of , meaning that is divisible by 5.
Next, we examine option D. We take the first few powers of added with :
We see that the odd powers of added with 1 are multiples of three. If we continue this pattern, will be divisible by .
Next, we examine option B. We see that has a units of digits of (Taking the units digit of the first few powers of two gives a pattern of ). Adding to , we get . Since has a units digit of , it is divisible by .
Lastly, we examine . Using the sum of cubes factorization , we have . Since ends with a , and , is a multiple of , which means it is divisible by .
Since we have eliminated every option except one, is not divisible by any prime less than .