AMC 10B 2022 (Problem 17)One of the following numbers is not divisible by any prime number less than 10. Which is it?(A) 2606−1\text{(A)}\;2^{606}-1(A)2606−1(B) 2606+1\text{(B)}\;2^{606}+1(B)2606+1(C) 2607−1\text{(C)}\;2^{607}-1(C)2607−1(D) 2607+1\text{(D)}\;2^{607}+1(D)2607+1(E) 2607+3607\text{(E)}\;2^{607}+3^{607}(E)2607+3607Related TopicsToolkit 13 — General difference of powersToolkit 14 — Sum of odd powerHints (7)Hint 1Refer to 13. General difference of powers and 14. Sum of odd powers.Hint 2a−b∣an−bn,n∈Z+a-b \mid a^n-b^n,\quad n\in\mathbb{Z}^+a−b∣an−bn,n∈Z+Hint 3a+b∣an+bn,n odda+b \mid a^n+b^n,\quad n \text{ odd}a+b∣an+bn,n oddHint 4Choice A: 2606−1=(22)303−1=4303−12^{606}-1=(2^2)^{303}-1=4^{303}-12606−1=(22)303−1=4303−1, so 4−1=3∣4303−14-1=3 \mid 4^{303}-14−1=3∣4303−1.Hint 5Choice B: 2606+1=4303+12^{606}+1=4^{303}+12606+1=4303+1, so 4+1=5∣4303+14+1=5 \mid 4^{303}+14+1=5∣4303+1.Hint 6Choice D: 2+1=3∣2607+12+1=3 \mid 2^{607}+12+1=3∣2607+1.Hint 7Choice E: 2+3=5∣2607+36072+3=5 \mid 2^{607}+3^{607}2+3=5∣2607+3607.Final Answer(C) 2607−12^{607}-12607−1