Which number will completely divide 4^61 + 4^62 + 4^63 + 4^64?

Given data

• Sum: 4^61 + 4^62 + 4^63 + 4^64

Concept / Approach

• Factor out the lowest power. Recognize a geometric sum.

Step-by-step

4^61 + 4^62 + 4^63 + 4^64 = 4^61(1 + 4 + 16 + 64)Inside sum = 85 = 5 × 17Thus the whole expression = 4^61 × 85It is divisible by 85. The other options 51 = 3 × 17 and 57 = 3 × 19 and 95 = 5 × 19 require a factor 3 or 19 which is not present.

Common pitfalls

• Missing the factorization and attempting long division.

Final Answer

Divisible by 85.