Question

If two positive integers x and y are expressible in terms of primes as x = p square q square and y = p cube q , what can you about their L.C.M and H.C.F ?Is l.c.m a multiple of h.c.f explain?

Answer 1

If y=p³q³ then their HCF will be X=p²q² and there LCM will be p³q⁴ or p⁴q³ whichever is less.

And yes lcm is multiple of hcf

Answer 2

Given :

x & y are two positive integers

x = p²q²

y = p³q

where p & q are prime numbers

To Find :

LCM & HCF of x & y

Solution:

• LCM is lowest common multiple & HCF is Highest common factor .

• FOR HCF

x = p×p×q×q

y = p×p×p×q

where p&q are prime factors of both x&y

•HCF of two Numbers is the maximum number which after division gives remainder as Zero

•So, maximum common factors in x & y are p×p×q

•So, HCF i.e. Highest common factor of x & y is p²q ______(1)

•For LCM

x = p×p×q×q

y = p×p×p×q

•where p&q are prime factors of both x&y

•LCM of two numbers is the lowest multiple of x&y which is same or equal.

•If, x is multiplied by p & y is multiplied by q then both will become equal to p³q²

•So, LCM i.e. lowest common multiple of x & y is p³q²

•From (1) we get ,

HCF of x&y = p²q &

LCM of x&y = p³q²

•Clearly , HCF ×pq = LCM

•hence , LCM i.e. lowest common multiple is multiple of HCF i.e. Highest common factor