2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 17 Medium

One of the following numbers is not divisible by any prime number less than 10. Which is it?

  • A.

    2^{606}-1

  • B.

    2^{606}+1

  • C.

    2^{607}-1

  • D.

    2^{607}+1

  • E.

    2^{607}+3^{607}

Answer:C

A. 2^{606} \equiv 4^{303} \equiv 1(\bmod 3) \Rightarrow 3 \mid 2^{606}-1.

B. 2^{606} \equiv 4^{203} \equiv -1(\bmod 5) \Rightarrow 5 \mid 2^{606}+1.

D. 2^{607} \equiv 4^{303} \times 2 \equiv 2(\bmod 3) \Rightarrow 3 \mid 2^{607}+1.

E. 2^{607}+3^{607} \equiv 2^{607}+(-2)^{607} \equiv 0(\bmod 5) \Rightarrow 5\mid 2^{607}+3^{607}.

So the answer is C.

Related Problems
Problem 15 2022 AMC 10 B
Problem 16 2022 AMC 10 B
Problem 18 2022 AMC 10 B
Problem 19 2022 AMC 10 B
Think Academy Powered by ChatGPT