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What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive 13 Apr 2019, 13:48
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75% (hard)

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55% (02:29) correct 45% (02:09) wrong based on 208 sessions

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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.
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D
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What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive 13 Apr 2019, 23:06
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Manang
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.
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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
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What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive 07 Jul 2019, 12:15
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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.
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What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive 01 Jun 2020, 03:17
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\(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.\\
\\
Show Spoiler
Hence, the correct answer is option D.
\\
\)
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What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive 13 Sep 2021, 21:01
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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


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What is the HCF of P and Q if 3P Q 5 = 0 and P and Q are positive 26 Oct 2023, 07:12
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