Basic Mathematics & Logarithm
Mathematical Induction
nta_abhyas_2025
Grade None
Question:
If $x$ and $y$ are two distinct integers and $n$ is a natural number then $x^n - y^n$ is divisible by
x^n + y^n
xy(x^n - y^n)
x + y
x - y
Step-by-Step Solution
Key Concept: The expression $x^n - y^n$ always has $(x-y)$ as a factor by the factorization of difference of powers
Let $P(n): x^n - y^n$ where $n \in \mathbb{N}$. Then $P(1): x - y$ is divisible by $x - y$. $P(2): x^2 - y^2$ is divisible by $x - y$. $P(3): x^3 - y^3$ is divisible by $x - y$. Hence $x^n - y^n$ is divisible by $x - y$.
Correct Answer: 3