Binomial Theorem
Fractional Part Using Binomial Expansion
nta_pyq_2023_apr
Grade 11
Question:
Fractional part of the number $\dfrac{4^{2022}}{15}$ is equal to
Step-by-Step Solution
Key Concept: Write $4^{2022}=16^{1011}=(15+1)^{1011}$. Expand and observe all terms except the last are divisible by 15.
$4^{2022}=(15+1)^{1011}\equiv 1\pmod{15}$. Fractional part $=\frac{1}{15}$.
Correct Answer: 4