The remainder when 3²⁰²² is divided by 5 is :

Solution

Maths sir · 4 steps

Write 3²⁰²² as 9¹⁰¹¹ = (10-1)¹⁰¹¹, and every term but the last is a multiple of 10.

1. Rewrite the power. Since 3² = 9, we have 3²⁰²² = 9¹⁰¹¹ = (10-1)¹⁰¹¹. This puts the base next to a multiple of 5, which is what the binomial expansion needs.

   (10-1)¹⁰¹¹ = C(1011, 0)10¹⁰¹¹ - C(1011, 1)10¹⁰¹⁰ + … + C(1011, 1010)10 - 1

2. Look at the terms. Every term except the last has a factor of 10. The last term is (-1)¹⁰¹¹ = -1, because 1011 is odd. So 3²⁰²² = 10m - 1 for some natural number m.
3. Fix the negative. A remainder cannot be negative, so rewrite it as 10m - 1 = 10m - 5 + 4 = 5(2m - 1) + 4. Here 0 ≤ 4 < 5, so 4 is the true remainder.
4. So. The remainder when 3²⁰²² is divided by 5 is 4, which is option (d).

The idea to keep. For remainders of big powers, write the base as (multiple of divisor ± 1). If you get a -1 left over, add the divisor to make it a proper remainder.

Answer from the NTA final answer key. Solution by PrepWiser.