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Difference between revisions of "2019 Mock AMC 10B Problems/Problem 22"
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==Solution==
 
==Solution==
  
−
S15^n - 7^n =\equiv 7^n - 7^n \equiv 0<math> </math>\text{mod}<math> </math>8<math> for all integer </math>n<math>. Therefore, </math>S = \boxed{0, 8, 16, 24,...,248\}<math>. Since the sum of the elements in </math>S<math> is </math>8 + 16 + 24 + ... + 248 = 8(1 + 2 + 3 + ... + 31) = 8 \cdot \frac{31(31 + 1)}{2} = 2^7 \cdot 31<math>, so the answer is </math>2 + 7 + 31 = \boxed{\text{(A)} 40}$.
+
S15^n - 7^n =\equiv 7^n - 7^n \equiv 0<math> </math>\text{mod}<math> </math>8<math> for all integer </math>n<math>. Therefore, </math>S = \{0, 8, 16, 24,...,248\}<math>. Since the sum of the elements in </math>S<math> is </math>8 + 16 + 24 + ... + 248 = 8(1 + 2 + 3 + ... + 31) = 8 \cdot \frac{31(31 + 1)}{2} = 2^7 \cdot 31<math>, so the answer is </math>2 + 7 + 31 = \boxed{\text{(A)} 40}$.

Revision as of 20:25, 3 November 2019

Problem

Let $S = \{r_1, r_2, r_3, ..., r_{\mu}\}$ be the set of all possible remainders when $15^{n} - 7^{n}$ is divided by $256$, where $n$ is a positive integer and $\mu$ is the number of elements in $S$. The sum $r_1 + r_2 + r_3 + ... + r_{\mu}$ can be expressed as\[p^qr,\]where $p, q, r$ are positive integers and $p$ and $r$ are as small as possible. Find $p+q+r$.

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$for all integer$n$. Therefore,$S = \{0, 8, 16, 24,...,248\}$. Since the sum of the elements in$S$is$8 + 16 + 24 + ... + 248 = 8(1 + 2 + 3 + ... + 31) = 8 \cdot \frac{31(31 + 1)}{2} = 2^7 \cdot 31$, so the answer is$2 + 7 + 31 = \boxed{\text{(A)} 40}$.

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