Question

If $aN=\left\{ax:x\in N\right\}$ and $bN\cap cN=dN$ where $b,c\in N$ are relatively prime then

• A. $d=bc$
• B. $c=bd$
• C. $b=cd$
• D. $a=bd$

Answer 1

The correct option is A

$d=bc$

Explanation for the correct option:\

Intersection of set:

Given,

$bN=\left\{bx:x\in N\right\}$

$cN=\left\{cx:x\in N\right\}$

$\therefore bN\cap cN=$ {$x:x$ is multiple of $b$ and $c$ both}

$=$ { $x:x$ is multiple of l.c.m. of $b$ and $c$}

$=$ { $x:x$ is multiple of $bc$}

$\begin{array}{rcl}\therefore bN\cap cN& =& \left\{bcx:x\in N\right\}\\ & =& dN\end{array}$ $\left[\because bN\cap cN=dN\right]$

$\mathbf{\therefore }\mathit{d}\mathbf{}\mathbf{=}\mathbf{}\mathit{b}\mathit{c}$

Hence, Option ‘A’ is Correct.