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amanvermagmat
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P = 2^a * 3^b, while Q = 3^c * 5^d, where a,b,c,d are all non-negative 25 Apr 2018, 11:05
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95% (hard)

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32% (02:02) correct 68% (01:45) wrong based on 74 sessions

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P = 2^a * 3^b, while Q = 3^c * 5^d, where a, b, c, d are all non-negative integers. What is the Greatest Common Divisor of P and Q?

(1) Lowest Common Multiple of P and Q is (2^3 * 3^2 * 5^4).

(2) a, b, c, d are all distinct from each other.
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E
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P = 2^a * 3^b, while Q = 3^c * 5^d, where a,b,c,d are all non-negative 25 Apr 2018, 11:50
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amanvermagmat
P = 2^a * 3^b, while Q = 3^c * 5^d, where a, b, c, d are all non-negative integers. What is the Greatest Common Divisor of P and Q?

(1) Lowest Common Multiple of P and Q is (2^3 * 3^2 * 5^4).

(2) a, b, c, d are all distinct from each other.

as between P & Q only common prime factor is 3, hence to find the GCD, we need to know the powers of 3 i.e. the value of \(b\) & \(c\)

Statement 1: from this we know that one of the powers of 3 is 2 but other power could be 2, 1, or 0. Hence the GCD could be \(3^2\), \(3\) or \(1\). Insufficient

Statement 2: no values can be found out from this statement. Insufficient

Combining 1 & 2: one of the power of 3 is 2 but the other powers can be 1 or 0. Hence the GCD can be 3 or 1. Insufficient

Option E
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P = 2^a * 3^b, while Q = 3^c * 5^d, where a,b,c,d are all non-negative 21 Feb 2020, 02:22
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amanvermagmat
P = 2^a * 3^b, while Q = 3^c * 5^d, where a, b, c, d are all non-negative integers. What is the Greatest Common Divisor of P and Q?

(1) Lowest Common Multiple of P and Q is (2^3 * 3^2 * 5^4).

(2) a, b, c, d are all distinct from each other.

Solution


Step 1: Analyse Question Stem


    • \(P = 2^a*3^b\)
    • \(Q = 3^c*5^d\)
      o \(a, b, c,\) and \(d\) are non-negative integers.
We need to find the GCD of P and Q.
    • 3 is the only prime factor which common to both P and Q, So there will be 3 cases as shown below:
    • If \(b>c\), then \(GCD(P, Q) = 3^c\), we need the value of c.
    • If \(b < c\), then \(GCD(P, Q) =3^b\), we need the value of b.
    • If \(b = c\), then \(GCD(P, Q)=3^b=3^c\), we need the value of either b or c
Now, To find the GDC of P and Q, we need to know the relation between b and c as well as there value

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE


Statement 1: \(LCM(P, Q) -2^3*3^2*5^4.\)
Here, we don't know whether \(3^2\) is the value \(3^b\) or \(3^c\). Let's consider all the cases
    • If \(b > c\), then \(b = 2\)
      o c can be 0 or 1.
         If \(c =0\), then \(GCD(P, Q) = 3^0=1\)
         If \(c = 1\), then \(GCD(P, Q)=3^1=3\)
    • If \(b < c\), then \(c = 2\)
      o b can be 0 or 1.
         If \(b = 0\), then \(GCD(P, Q) = 3^0=1\)
         If \(b = 1\), then \(GCD(P, Q)=3^1=3\)
    • If \(b= c\), then \(GCD(P, Q) = 3^2 = 9\)
We are getting different possible values of GCD.
Hence, statement 1 is not sufficient, we can eliminate answer options A and D.
Statement 2: \(a, b, c,\) and \(d\) are distinct integers.
    • We cannot infer anything about the value of \(b\) or \(c\).
Hence, statement b is also not sufficient, we can eliminate answer options B.

Step 3: Analyse Statements by combining.


From statement 1:\( GCD(P, Q)\) can be \(1, 3\), or \(9\)
From statement 2: \(a, b, c, d\) are distinct integers.
By combining both the statements, we get
    • \(b\) is not equal to \(c\)
      o \(GCD(P, Q)\) is not equal to \(9\)
    • Still, there are two possible values of \(GCD(P, Q)\) i.e., \(1\) and \(3\).
Hence, the correct answer is Option E
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P = 2^a * 3^b, while Q = 3^c * 5^d, where a,b,c,d are all non-negative 21 Feb 2020, 10:05
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amanvermagmat
P = 2^a * 3^b, while Q = 3^c * 5^d, where a, b, c, d are all non-negative integers. What is the Greatest Common Divisor of P and Q?

(1) Lowest Common Multiple of P and Q is (2^3 * 3^2 * 5^4).

(2) a, b, c, d are all distinct from each other.

We have \(P = 2^a * 3^b\) and \(Q = 3^c * 5^d\). Their only common factor of 3, so their common divisor has to be a factor of 3. The greatest common divisor will be depending on b or c, whichever is smaller. Hence we would like to know min(b, c).

Statement 1:
Their LCM is given, focus on the factor of 3. 3 has a power of two, this tells us max(b, c) is 2. However we may have
(i) b = 1, c = 2. Min(b, c) = 1.
(ii) b = 2, c = 2. Min(b, c) = 2.

We cannot confirm an answer, insufficient.

Statement 2:
Insufficient.

Combined:
Note the condition on b and c was they are nonnegative. Hence b or c can be 0, 1, or 2 with statement 1. Thus even combined with statement 2 we can have:
(i) b = 0, c = 2. Min(b, c) = 0.
(ii) b = 1, c = 2. Min(b, c) = 1.

Insufficient.

Ans: E
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