Question

Define $x\%a$ as the remainder obtained when $x$ is divided by $a$.
Let $f:Z\to R$ be defined as $f\left(x\right)=x\%k$, where $kϵN$. If Y is the range of $f\left(x\right)$, then

• A. $Y=R$
• B. $Y=[0,k]$
• C. $Y=N\cup \left\{0\right\}$
• D. $n\left(Y\right)=k$

Answer 1

The correct option is D

$n\left(Y\right)=k$

Given x is an integer and k is a natural number, x can be expressed as
$x=q.k+r$, where $q,rϵZ$ and $0\le r<k$.
i.e. the remainder can be any value from 0 to k-1
$\Rightarrow rϵ\{0,1,2.......k-1\}\Rightarrow f\left(x\right)ϵ\{0,1,2........k-1\}\Rightarrow Y=\{0,1,2,..........k-1\}\Rightarrow n\left(Y\right)=k$