Question

The fractional part of the number \(\tfrac{4^{2022}}{15}\) is equal to:

Answer 1

Given:
Find fractional part of (4²⁰²² / 15)

Step 1: Idea
Fractional part of (N / 15) = (remainder of N ÷ 15) / 15

Step 2: Find 4²⁰²² mod 15
4¹ = 4
4² = 16 ≡ 1 (mod 15)

So pattern:
4^2k ≡ 1 (mod 15)

Since 2022 is even:
4²⁰²² ≡ 1 (mod 15)

Step 3: Fractional part
\[ \frac{4^{2022}}{15} = \text{integer} + \frac{1}{15} \]

Fractional part = 1/15