Question

If the integers $m$ and $n$ are chosen at random between 1 and 100, then the probability that a number of the form $7^m + 7^n$ is divisible by 5, equals

• A. $\frac{1}{4}$
• B. $\frac{1}{7}$
• C. $\frac{1}{8}$
• D. $\frac{1}{49}$

Hint:

Always check cycle of powers in modulo problems.

Answer 1

The Correct Option is A

Concept: Use cyclicity in modular arithmetic.

Step 1: Reduce modulo 5.
\[ 7 \equiv 2 \pmod{5} \Rightarrow 7^m \equiv 2^m \]

Step 2: Find cycle of $2^m$.
\[ 2^1=2,\ 2^2=4,\ 2^3=3,\ 2^4=1 \] Cycle length = 4

Step 3: Condition for divisibility.
\[ 2^m + 2^n \equiv 0 \pmod{5} \] Possible pairs satisfy complementary residues.

Step 4: Count favorable cases.
Probability = $\frac{1}{4}$ Conclusion:
Answer = $\frac{1}{4}$