The least common multiple of positive integers p and q is 630. if the

\(630 = 3 * 3*2*5*7\)

Possible values of P and Q keeping in mind that HCF cannot be anything other than \(3 * 3 *2\)

Case 1:

P \(= 3 * 3*2*5*7\)
Q \(= 3 * 3*2\)


Case 2:

P \(= 3 * 3*2\)
Q \(= 3 * 3*2*5*7\)

Case 3:

P \(= 3 * 3*2*5\)
Q \(= 3 * 3*2*7\)

All of the above values of P and Q are the ONLY cases that will give a LCM of \(630\) and HCF of \(18\), we just need to find the correct values of P and Q to arrive at P-Q.

(1) P is a multiple of \(35\) : This is only possible with case \(1\) hence we know P \(= 3 * 3*2*5*7\) and Q \(= 3 * 3*2\)

We can find P-Q. SUFF.


(2) The GCD of Q and \(35\) is \(1\) :

This tells us that \(7\) and \(5\) are NOT part of Q hence \(7 \)and \(5\) must be part of P.
Hence P \(= 3 * 3*2*5*7\) and Q\(= 3 * 3*2\)

We can find P-Q. SUFF.

Ans D

Hope it's clear.­