If P and Q are positive integers, is P*Q divisible by 63?

If P and Q are positive integers, is P*Q divisible by 63?

63 = 3*3*7. So we need that P*Q has 3*3*7 as factors.

(1) The greatest common factor of P and Q is 6.
Insufficient. GCF is 6 (3*2). So we know that P and Q can be factored as P = 2*3*x and Q=2*3*y Here we see that x and y can be equal to whatever we want so if the factors behind x and y are 3 and 7 (like for example P=2*3*7*2 Q=2*3*7) we can answer that P*Q is divisible. Otherwise (for example P=2*3*11 Q=2*3 ==> we don't have 7 here) the product is not divisible.

(2) The least common multiple of P and Q is 1260.
Sufficient. 1260=2*2*3*3*5*7. It means that factors of P and Q contain these numbers. We have in these factors 3*3*7 ==> The product is divisible. (For example P=2*2*3*5 Q=2*3*3*7)

Answer B

Asked: If P and Q are positive integers, is P*Q divisible by 63?

63 = 3*3*7 = 3ˆ2*7
We need to confirm that product P*Q contains these factors.

(1) The greatest common factor of P and Q is 6.
P = 2*3*x & Q=2*3*y
But we are not sure that x*y contains 3*7 or not.
NOT SUFFICIENT

(2) The least common multiple of P and Q is 1260.
1260 = 2*2*5*3*3*7 = 2ˆ2*3ˆ2*5*7
Since P*Q contains 3ˆ2*7
P*Q is divisible by 63
SUFFICIENT

IMO B
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6 multiplied by 2,3,4,5,6 and 7 all have solutions that have 6 as GCF. So statement 1 is insufficient.

42 × 30 = 1260. Therefore P and Q can be 42 and 30 with 1260 as LCM. 1260 is divisible by 63. So statement 2 is sufficient.

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We have the formula for any two positive integers a and b: \(LCM = \frac{a*b}{GCF}\).

Statement 1:
Insufficient as we can't confirm if P and Q have a factor of 7 or 9 to build 63. (Something like GCF is 21 would be sufficient, however, we have 3 in both and 7 in both in that case to build 63 = 3*3*7.).

Statement 2:
Using the formula above, \(LCM = \frac{P*Q}{GCF}\). GCF is at least 1, so we have \(P*Q = 1260 * GCF\) which is a multiple of 63. Sufficient.

Ans: B
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IMO ANSWER B

If P and Q are positive integers, is P*Q divisible by 63?

(1) The greatest common factor of P and Q is 6.
The GCF of P and Q is 6. Here P and Q can take any values - Insufficient.

(2) The least common multiple of P and Q is 1260.
P*Q = 1260
Prime factorization of 1260 gives us = \(2^2 * 3^3 * 5 * 7 \)

So P * Q has \(3^2 * 7\) in them =, so P*Q is divisible by 63.
Option B Sufficient

Ans B
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IMO B


If P and Q are positive integers, is P*Q divisible by 63?

(1) The greatest common factor of P and Q is 6.
Can take varying examples that will satisfy/not satisfy the above conditions.
Hence not sufficient.

(2) The least common multiple of P and Q is 1260.
Upon prime factorization of 1260 - 7 * 180
Here, we see that, 7 and 9, both are the factors of the LCM hence the number will be divisible by 63 ultimately.

(1) insf
gcf pq = 6 = 2,3
no info pq has 7

(2) sufi
lcm(pq)=1260=126*10=21*6*10
this includes 3^2 and 7, div by 63

(B)

(1) PQ can take several values whose GCF is \(6\);

For example, \(30*12= 360\); GCF is \(6\) but \(360 \) is not divisible by \(63\)

On the other hand, \(30*42=1260\); GCF is 6; and 1260 is divisible by \(63\)

Insufficient.

(1) As \(LCM \) is \(1260 \)that means \(PQ=1260\), which is a multiple of \(63\). Sufficient.
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