If aN = ax: x ∈ N and bN ∩ cN = dN, where b, c ∈ N are relatively prime, then

Question

If aN = ax: x ∈ N and bN ∩ cN = dN, where b, c ∈ N are relatively prime, then (A) d = bc (B) c = bd (C) b = cd (D) None of these

Tardigrade Admin · Accepted Answer

bN = bx: x ∈ N cN = cx: x ∈ N ∴ bN ∩ 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 ∴ l.c.m. of b and c = bc] ∴ bN ∩ cN = bc x: x ∈ N = dN (Given) ∴ d = bc.

Q. If $a N = a x : x \in N$ and $b N \cap c N = d N$ , where $b, c \in N$ are relatively prime, then

$d = b c$

$c = b d$

$b = c d$

None of these

Q. If aN=ax:x∈N and bN∩cN=dN, where b,c∈N are relatively prime, then

d=bc

c=bd

b=cd

None of these

bN = {bx:x∈N} cN = {cx:x∈N} ∴bN∩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 ∴ l.c.m. of b and c=bc] ∴bN∩cN=bcx:x∈N=dN (Given) ∴d=bc.

Q. If $a N = a x : x \in N$ and $b N \cap c N = d N$ , where $b, c \in N$ are relatively prime, then

$d = b c$

$c = b d$

$b = c d$