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OFFICIAL EXPLANATIONStep 1: Pre-ThinkingIn 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 StatementsAnalysing 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.
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