Question

Which of the following divides $3^{45 + 3^{46} + 3^{47} + 3^{48} + 3^{49}$?

• A. \(3 \)
• B. \(5 \)
• C. \(2 \)
• D. \(11 \)
• E. \(4 \)

Hint:

When powers are in sequence: - Factor out the smallest power - Convert into geometric series for easy simplification

Answer 1

The Correct Option is A

Concept: Factor out the smallest power: \[ a^m + a^{m+1} + \cdots = a^m(1 + a + a^2 + \cdots) \]
Step 1: Factor the expression.
\[ 3^{45} + 3^{46} + 3^{47} + 3^{48} + 3^{49} = 3^{45}(1 + 3 + 3^2 + 3^3 + 3^4) \]

Step 2: Simplify bracket.
\[ 1 + 3 + 9 + 27 + 81 = 121 \] \[ = 3^{45} \times 121 \]

Step 3: Analyze divisibility.
\[ 121 = 11^2 \] So: \[ 3^{45} \times 121 \] is divisible by: \[ 3 \text{ and } 11 \]

Step 4: Check options.
Among given options, the valid answer is:3