JEE Main 2020 · 6 September · morning · Single correct · +4 / −1

If {p} denotes the fractional part of the number p, then
{ 3²⁰⁰/8 }, is equal to :

  1. (a) 5/8
  2. (b) 7/8
  3. (c) 1/8
  4. (d) 3/8
Answer and solution

Answer (c) 1/8

Solution

Maths sir · 3 steps

Turn 3²⁰⁰ into 9¹⁰⁰, write 9 as 8 + 1, and every term except the last is a multiple of 8.

  1. Make 8 appear. The fractional part of N/8 depends only on the remainder when N is divided by 8. Write 3²⁰⁰ = (3²)¹⁰⁰ = 9¹⁰⁰ = (8 + 1)¹⁰⁰.

    (8+1)¹⁰⁰ = 8¹⁰⁰ + ¹⁰⁰C₁ 8⁹⁹ + … + ¹⁰⁰C₉₉ 8 + 1

  2. Read off remainder. Every term except the last has at least one factor of 8, so their sum is 8k for some integer k. So 3²⁰⁰ = 8k + 1, and 3²⁰⁰/8 = k + 1/8.
  3. So. The integer part is k and the fractional part is 1/8, which is option (c).

The idea to keep. For a remainder or fractional-part question, rewrite the base as (multiple of the divisor) ± 1. After the binomial expansion, only the last term survives.

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

Study Binomial Theorem

Asked about 200 times in JEE Main since 2020. This question is from 2020.