QB_PH_GA_011: Quantitative Aptitude

What is the remainder when $7^{104}$ is divided by $25$?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Modular Arithmetic Technique**: $7^1 = 7 \equiv 7 \pmod{25}$ $7^2 = 49 = 50 - 1 \equiv -1 \pmod{25}$ 2. **Exponentiation of Quadratic Residue**: $7^4 = (7^2)^2 \equiv (-1)^2 \equiv 1 \pmod{25}$ 3. **Decomposition of Power**: $104 = 4 \times 26$ $7^{104} = (7^4)^{26} \equiv (1)^{26} \equiv 1 \pmod{25}$ 4. **Official Answer**: 1.