Question

Which one of the following numbers will completely divide \(4^{61} + 4^{62} + 4^{63} + 4^{64}\)?

• A. 3
• B. 10
• C. 11
• D. 13

Hint:

Factor out 4^61 to get 4^61 x 85, then check which option divides 2^122 x 5 x 17.

Answer 1

The Correct Option is B

Step 1: Factor out the common term.
The expression is \(4^{61} + 4^{62} + 4^{63} + 4^{64}\). Every term has \(4^{61}\) as a common factor, so factor it out: \(4^{61}(1 + 4 + 4^2 + 4^3)\).
Step 2: Evaluate the bracket.
\(1 + 4 + 16 + 64 = 85\). So the expression becomes \(4^{61} \times 85\).
Step 3: Break 85 into its prime factors.
\(85 = 5 \times 17\). So the full expression is \(4^{61} \times 5 \times 17\), which can also be written as \(2^{122} \times 5 \times 17\) since \(4 = 2^2\).
Step 4: Check which option divides this exactly.
The expression contains only the prime factors 2 (many times, from \(2^{122}\)), 5, and 17. It has no factor of 3, no factor of 11, and no factor of 13, so options (1) 3, (3) 11 and (4) 13 cannot divide it exactly.
Step 5: Confirm option (2).
Since the expression has both a factor of 2 and a factor of 5 (from \(2^{122} \times 5 \times 17\)), it certainly has a factor of \(2 \times 5 = 10\). So 10 divides the expression completely, and the correct option is (2).