Difference between revisions of "2019 Mock AMC 10B Problems/Problem 22"
(Created page with "==Problem== Let <math>S = \{r_1, r_2, r_3, ..., r_{\mu}\}</math> be the set of all possible remainders when <math>15^{n} - 7^{n}</math> is divided by <math>256</math>, where...") |
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Revision as of 20:23, 3 November 2019
Problem
Let 
be the set of all possible remainders when 
is divided by 
, where 
is a positive integer and 
is the number of elements in 
. The sum 
can be expressed as![\[p^qr,\]](http://latex.artofproblemsolving.com/5/7/e/57ef78a97f88f2c6b8fe060205ea30c4927b9520.png)
where 
are positive integers and 
and 
are as small as possible. Find 
.
Solution
S15^n - 7^n =\equiv 7^n - 7^n \equiv 0$$ (Error compiling LaTeX. Unknown error_msg)\text{mod}$$ (Error compiling LaTeX. Unknown error_msg)8
n
S = \boxed{0, 8, 16, 24,...,248\}
S
8 + 16 + 24 + ... + 248 = 8(1 + 2 + 3 + ... + 31) = 8 \cdot \frac{31(31 + 1)}{2} = 2^7 \cdot 31, so the answer is 
.
