2022 AMC 10B Problem 17

Use the fact that $a^n-b^n$ is divisible by $a-b.$ Choice A is $2^{606}-1=4^{303}-1,$ which is divisible by $4-1=3.$ Choice B is $2^{606}+1=4^{303}-(-1)^{303},$ which is divisible by $4-(-1)=5.$ Choice D is $2^{607}+1.$ Since $2^{606}-1$ is divisible by $3,$ multiplying by $2$ gives $2^{607}-2$ divisible by $3,$ so $2^{607}+1$ is divisible by $3.$ Choice E is $3^{607}+2^{607}=3^{607}-(-2)^{607},$ which is divisible by $3-(-2)=5.$ For choice C, $2^{607}-1$ is odd. Also $2^{607}\equiv2\pmod3,$ $2^{607}\equiv3\pmod5,$ and $2^{607}\equiv2\pmod7,$ so $2^{607}-1$ is not divisible by $3,5,$ or $7.$ Thus, our answer is C .