What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive

Question

What is the HCF of P and Q if 3P – Q – 5 = 0 and P and Q are positive integers?

(1) P is completely divisible by 5.
(2) Q is completely divisible by 5.

GMATinsight · Accepted Answer

Question : HCF of P and Q = ?

Given: 3P – Q – 5 = 0
i.e. 3P – Q = 5

Statement 1 : P is completely divisible by 5

Case 1: P = 5, then Q = 10 HCF = 5
Case 2: P = 10, then Q = 25 HCF = 5
Case 3: P = 15, then Q = 40 HCF = 5
Case 4: P = 20, then Q = 55 HCF = 5

i.e. HCF in each case is 5 i.e.

SUFFICIENT

Statement 2 : Q is completely divisible by 5

Case 1: P = 5, then Q = 10 HCF = 5
Case 2: P = 10, then Q = 25 HCF = 5
Case 3: P = 15, then Q = 40 HCF = 5
Case 4: P = 20, then Q = 55 HCF = 5

i.e. HCF in each case is 5 i.e.

SUFFICIENT

Answer: Option D

Chethan92 · Answer

3P – Q – 5 = 0

From S1:

P is completely divisible by 5 or P is a multiple of 5.
P can be 0,5,10,15,20,25.......
If P = 0, then Q = -5. Not possible as P and Q are positive integers.
Min value of P has to be 5.
P = 5, then Q = 10
P = 10, then Q = 25
P = 15, then Q = 40
and so on
HCF (P,Q) = 5.
SUFFICIENT.

From S2:

Q is completely divisible by 5 or Q is a multiple of 5.
Q can be 0,5,10,15,20,25.......
If Q = 0, then P = 5/3. Not possible as P and Q are positive integers.
Min value of Q has to be 5.
Same case as S1.
HCF (P,Q) = 5.
SUFFICIENT.

D is the answer.

coreyander · Answer

OFFICIAL EXPLANATION

Step 1: Pre-Thinking

In this question, we are given:

P and Q are positive integers.
3P – Q – 5 = 0.
We need to find:

The HCF of P and Q.
Although we don’t have any specific inference regarding the values of P and Q, we can say that:

P = ${1 \over 3}$ (Q + 5)
And similarly, Q = 3P – 5
With this understanding, let’s now analyse the individual statements.

Step 2: Analysing Individual Statements

Analysing Statement 1:

As per the information given in statement 1, P is completely divisible by 5.

Therefore, we can say P = 5k, where k is a positive integer.
As P = ${1 \over 3}$ (Q + 5), we can equate the values of P and say that

5k = ${1 \over 3}$ (Q + 5)
Or, 15k = Q + 5

Or, Q = 15k – 5 = 5 (3k – 1)

Hence, P = 5k and Q = 5 (3k – 1)

As 3k and (3k – 1) are two consecutive integers, they don’t share any common factor among them.

Or in other words, the HCF of 3k and (3k – 1) is always 1.

Hence, the HCF of P and Q is 5, as 5 is the only common factor between P and Q.

As we can determine the HCF of P and Q, we can conclude that statement 1 is sufficient to answer the question.

Analysing Statement 2:

As per the information given in statement 2, Q is completely divisible by 5.

Therefore, we can say Q = 5m, where m is a positive integer.
As Q = 3P – 5, we can equate the values of P and say that

5m = 3P – 5
Or, 3P = 5m + 5 = 5 (m + 1)

Or, P = ${5 \over 3}$ (m + 1)

Now, as per the question, P is an integer. Hence, we can say that, ${5 \over 3}$ (m + 1) must also be an integer.

Thus, (m + 1) must be a multiple of 3, since 5 is not divisible by 3.
If we assume (m + 1) = 3b, where b is a positive integer, then we can write m = 3b – 1.

Therefore, P = ${5 \over 3}$ (m + 1) = ${5 \over 3}$ (3b – 1 + 1) = 5b
And, Q = 5m = 5(3b – 1)
Now, we know that any two consecutive numbers don’t share any common factor among them.

Hence, 3b and (3b – 1) don’t have any common factor.

Also, 3b is a multiple of b, hence, 3b will have all the factors of b.
Therefore, b will also have no common factor with (3b – 1).
Or in other words, the HCF of b and (3b – 1) is always 1.
Hence, the HCF of P and Q is 5, as 5 is the only common factor between P and Q.
Therefore, statement 2 is sufficient to answer the question.

Step 3: Combining Both Statements
 
Since we got a unique answer from both statement 1 and statement 2 independently, this step is not required.

Hence, the correct answer is option D.

chetan2u · Answer

What is the HCF of P and Q if 3P – Q – 5 = 0 and P and Q are positive integers?

1) P is completely divisible by 5.
 Let P=5x, so 3*5x-Q-5=0 
Q=15x-5=5(3x-1).
Now x and 3x-1 will never have any factor other than 1 common. 
Hence 5(x) and 5(3x-1) will have only 1 and 5 as common factors. 
Answer is 5.
Sufficient

2) Q is completely divisible by 5.
 Let Q=5y, so 3P-5y-5=0 
3P=5y-5=5(y-1).
Now y and y-1 will never have any factor other than 1 common. 
Hence 5(y) and 5(y-1), that is 3P and Q, will have only 1 and 5 as common factors. 
3P can be divisible by 5, only when P is divisible by 5.
Answer is 5.
Sufficient

D

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What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive

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