Fractional part of the number 4²⁰²²/15 is equal to

Solution

Maths sir · 4 steps

Spot that 4² = 16 is one more than 15, and the whole huge power collapses to remainder 1.

1. Fractional part. If a positive integer N leaves remainder r on division by 15, then N/15 = q + r/15. So the fractional part of N/15 is r/15, and we only need the remainder of 4²⁰²² on division by 15.
2. Pair the fours. Since 2022 is even, 4²⁰²² = (4²)¹⁰¹¹ = 16¹⁰¹¹ = (15 + 1)¹⁰¹¹.

   (15+1)¹⁰¹¹ = 15¹⁰¹¹ + C(1011, 1)15¹⁰¹⁰ + … + C(1011, 1010)15 + 1 = 15k + 1

3. Read the remainder. Every term of the binomial expansion except the last has a factor of 15. So 4²⁰²² leaves remainder 1 on division by 15.
4. So. 4²⁰²²/15 = k + 1/15, and the fractional part is 1/15, option (c).

The idea to keep. For the fractional part of aⁿ/m, find the remainder of aⁿ on division by m. Write a power of a as m ± 1 and use the binomial theorem.

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