JEE Advanced Maths: Number Theory / Modular Arithmetic questions with solutions

Q1. When 2³⁵ * 3¹⁶ is divided by 11, what is the remainder?

1. 1
2. 3
3. 5
4. 8

Answer: 1

2¹⁰ ≡ 1 (mod 11) by Fermat's little theorem. 2³⁵ = (2¹⁰)³ * 2⁵ ≡ 1 * 32 = 32 ≡ 10 (mod 11). For 3: 3¹=3, 3²=9, 3³=27≡5, 3⁴≡15≡4, 3⁵≡12≡1 (mod 11). So order of 3 is 5. 3¹⁶ = 3^(15+1) = (3⁵)³ * 3 ≡ 1*3 = 3 (mod 11). Product: 2³⁵ * 3¹⁶ ≡ 10 * 3 = 30 ≡ 8 (mod 11). Remainder = 8.