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28 Sep 2021, 01:30
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D
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Difficulty:
95%
(hard)
Question Stats:
39% (02:08) correct
61%
(02:21)
wrong
based on 117
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The least common multiple of positive integers p and q is 630. if the greatest common factor of P and Q is 18, what is P - Q?
(1) P is a multiple of 35
(2) The GCD of Q and 35 is 1
(1) P is a multiple of 35
(2) The GCD of Q and 35 is 1
30 Sep 2021, 09:09
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Bunuel
\(630 = 3 * 3*2*5*7\)
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.
30 Sep 2021, 09:27
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Bunuel
We have the formula \(LCM = \frac{a*b}{GCF}\). In this case, we can find \(p*q = 630*18\).
Both p and q must have a GCF. Then we can break the product down into p*q = 35*18*18, where we need to distribute 35 across p or q. 35 has no common factors with 18, so distributing 35 will not affect the GCF. We must either split 35 into 5*7 or simply give 35 to one of p or q.
Statement 1:
This tells us p must be 35*18. Then q must be 18. Sufficient.
Statement 2:
Q does not have any factor of 35. Then P must accept the 35. p = 35*18 and q = 18. Sufficient.
Ans: D
