Question

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

• A. $d = bc$
• B. $c = bd$
• C. $b = cd$
• D. None of these

Answer 1

Answer: (A)

$d = bc$

Answer 2

bN = {$bx : x \in N$} cN = {$cx : x \in N$} $\therefore bN \cap cN =$ {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 b c} [given b and c are relatively prime $\therefore$ l.c.m. of b and $c = bc$] $\therefore bN \cap cN = {bc x : x \in N} = dN$ (Given) $\therefore d = bc.$