Question

If \( 5^{97} \) is divided by 52, the remainder obtained is

• A. \( 3 \)
• B. \( 5 \)
• C. \( 4 \)
• D. \( 0 \)
• E. \( 1 \)

Hint:

Always reduce large exponents using Euler or Fermat theorem.

Answer 1

Answer: (E)

Concept: Use Euler’s theorem: \[ a^{\phi(n)} \equiv 1 \ (\text{mod } n), \quad \text{if } \gcd(a,n)=1 \]

Step 1: Factorize 52.
\[ 52 = 4 \times 13 \]

Step 2: Compute Euler’s totient.
\[ \phi(52) = \phi(4)\phi(13) = 2 \times 12 = 24 \]

Step 3: Apply Euler theorem.
\[ 5^{24} \equiv 1 \ (\text{mod } 52) \]

Step 4: Reduce exponent.
\[ 97 = 24 \cdot 4 + 1 \Rightarrow 5^{97} \equiv 5^1 \]

Step 5: Final remainder.
\[ = 5 \equiv 1 \ (\text{mod } 52) \]