Question

What is the unit’s digit of the number (8pqr)⁶⁴ where p, q and r are the hundredth, tenth and units digits of the number?
Statement 1: The product of p and q is 12 
Statement 2: The product of q and r is 24 and r is greater than 4
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is C

To find the unit’s digit of the number \((8pqr)^{64}\), we need to know the unit's digit of the base, which is \(r\). The statements provided will help determine the values of \(p\), \(q\), and \(r\).

Analyzing Statement 1: The product of \(p\) and \(q\) is 12.

• Possible pairs for \((p, q)\) satisfying \(p \times q = 12\) are \((1, 12)\), \((2, 6)\), \((3, 4)\), \((4, 3)\), \((6, 2)\), and \((12, 1)\). However, since \(p\) and \(q\) are digits, possible pairs become \((2, 6)\) and \((3, 4)\).

However, this doesn't help in determining \(r\) as it could be any digit from 0 to 9. Thus, Statement 1 alone is insufficient.

Analyzing Statement 2: The product of \(q\) and \(r\) is 24 and \(r\) is greater than 4.

• Possible pairs for \((q, r)\) satisfying \(q \times r = 24\) where \(r > 4\) are \((6, 4)\), \((4, 6)\), \((8, 3)\), and \((3, 8)\). But considering \(r > 4\), thus possible combinations are \((6, 4)\) and `(3, 8)` ignoring as per initial choices deductions pairing remains (8, 3), (3, 8).

This statement is also insufficient by itself because knowing \(r\) depends on \(q\) and there's ambiguity without knowing \(q\). Therefore, Statement 2 alone is insufficient.

Combining Statements 1 and 2:

• From Statement 1, potential values of \((p, q)\) are \((2, 6)\) and \((3, 4)\).
• From Statement 2, potential values of \((q, r)\) where \(r >4\) are \((6, 4)\) -> Means (p, q) -> (2, 6) combination is eligible.
• Now \(q = 6\) and \(r = 4\), as we have only (2, 6) fitting scenario. Therefore, the unit digit of \(r\) is 4.

Now, the unit's digit of \((8pqr)^{64}\) is the unit's digit of \(4^{64}\).

The units digit of powers of 4 follow this pattern: 4, 6, 4, 6, ...

• \(64 \div 2 = 32 \), a multiple of 2, hence the unit's digit is 6.

Thus, both statements together help identify \(r\) as 4 and hence lead to the final answer. Consequently, the correct option is that both the statements together are needed to answer the question.