For what values of m in mathbb{N},does the di | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) We want to find $m \in \mathbb{N}$ such that $(x+y)$ divides $(x^m+y^m)$.Let $P(m)$ be the statement that $(x+y) \mid (x^m+y^m)$.For $m=1$: $x^1+y

Vedclass · Accepted Answer

Odd numbers

Vedclass · Answer

(B) We want to find $m \in \mathbb{N}$ such that $(x+y)$ divides $(x^m+y^m)$.Let $P(m)$ be the statement that $(x+y) \mid (x^m+y^m)$.For $m=1$: $x^1+y^1 = x+y$,which is clearly divisible by $x+y$. Thus,$P(1)$ is true.For $m=2$: $x^2+y^2$ is not generally divisible by $x+y$ (e.g.,if $x=1, y=1$,$2$ is divisible by $2$,but if $x=2, y=1$,$5$ is not divisible by $3$). Thus,$P(2)$ is false.For $m=3$: $x^3+y^3 = (x+y)(x^2-xy+y^2)$,which is divisible by $x+y$. Thus,$P(3)$ is true.In general,$x^m+y^m$ is divisible by $x+y$ if and only if $m$ is an odd natural number.Therefore,the divisibility holds for all odd numbers $m \in \mathbb{N}$.

For what values of $m \in \mathbb{N}$,does the divisibility $(x+y) \mid (x^m+y^m)$ hold?

Similar Questions

If $X = \{ 8^n - 7n - 1 : n \in N \}$ and $Y = \{ 49(n - 1) : n \in N \}$,then

The remainder when $64^{32^{32}}$ is divided by $9$ is equal to .........................

The remainder,when $(15^{23} + 23^{23})$ is divided by $19$,is

If $2 \cdot 4^{2k+1} + 3^{3k+1} = 11t$ and $2 \cdot 4^{2k+3} + 3^{3k+4} = 11(pt + 3^q)$,where $k, t \in Z^{+}$,then $(p, q) =$

For any integer $n \geq 1$,the remainder when the expression $n^5-5n^3+4n+139$ is divided by $120$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module