Difference between revisions of "2022 AMC 10B Problems/Problem 17"

Revision as of 16:25, 17 November 2022

Problem

One of the following numbers is not divisible by any prime number less than 10. Which is it? $\textbf{(A) } 2^{606}-1 \qquad\textbf{(B) } 2^{606}+1 \qquad\textbf{(C) } 2^{607}-1 \qquad\textbf{(D) } 2^{607}+1\qquad\textbf{(E) } 2^{607}+3^{607}$

Solution

For A, modulo 3, $\begin{align*} 2^{606} - 1 & \equiv (-1)^{606} - 1 \\ & \equiv 1 - 1 \\ & \equiv 0 . \end{align*}$

Thus, $2^{606} - 1$ is divisible by 3.

For B, modulo 5, $\begin{align*} 2^{606} + 1 & \equiv 2^{{\rm Rem} ( 606, \phi(5) )} + 1 \\ & \equiv 2^{{\rm Rem} ( 606, 4 )} + 1 \\ & \equiv 2^2 + 1 \\ & \equiv 0 . \end{align*}$

Thus, $2^{606} + 1$ is divisible by 5.

For D, modulo 3, $\begin{align*} 2^{607} + 1 & \equiv (-1)^{607} + 1 \\ & \equiv - 1 + 1 \\ & \equiv 0 . \end{align*}$

Thus, $2^{607} + 1$ is divisible by 3.

For E, module 5, $\begin{align*} 2^{607} + 3^{607} & \equiv 2^{607} + (-2)^{607} \\ & \equiv 2^{607} - 2^{607} \\ & \equiv 0 . \end{align*}$

Thus, $2^{607} + 3^{607}$ is divisible by 5.

Therefore, the answer is $\boxed{\textbf{(C) }2^{607} - 1}$ .

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

~MrThinker (LaTeX Error)

Solution 2 (Elimination)

Mersenne Primes are primes of the form $2^n-1$ , where $n$ is prime. Using the process of elimination, we can eliminate every option except for $\textbf{(A)}$ and $\textbf{(C)}$ . Clearly, $606$ isn't prime, so the answer must be $\boxed{\textbf{(C) }2^{607}-1}$ .

Video Solution

https://youtu.be/YF3HPVcVGZk

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

2022 AMC 10B (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 AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 57: / Line 57: @@
 ~MrThinker (LaTeX Error)
+==Solution 2 (Elimination)==
+Mersenne Primes are primes of the form <math>2^n-1</math>, where <math>n</math> is prime. Using the process of elimination, we can eliminate every option except for <math>\textbf{(A)}</math> and <math>\textbf{(C)}</math>. Clearly, <math>606</math> isn't prime, so the answer must be <math>\boxed{\textbf{(C) }2^{607}-1}</math>.
 ==Video Solution==

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