Question

LCM of two number x and y is 720 and the LCM of numbers 12x and 5y is also 720 .the no y is ;
a 180
b 144
c 120
d 90

Answer 1

1. Given two positive integers x and y, the relation

x×y=g.c.d(x,y)×lcm(x,y)xygcdxylcmxy always holds.

2. If g.c.d(x,y)=dgcdxyd, then g.c.d(ax,by)=daxbyd if g.c.d(a,b)=1gcdab1

Suppose g.c.d(x,y)=dgcdxyd then gcd(12x,5y)=dgcd12x5yd

Then, xy=720dxy720d

and, (12x)×(5y)=720d12x5y720d

⟹60xy=720d60xy720d

⟹60(720d)=720d60720d720d

⟹d=0d0, which is absurd since g.c.d(x,y)≠0gcdxy0

Therefore there doesn't exist any such positive integers xx and yy for l.c.mlcm(x,y)=720xy720 and l.c.m(12x,5y)=720.lcm12x5y720.