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\

• 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 determine the unit's digit of the number \((8pqr)^{64}\), we need to identify the unit's digit of 8pqr itself. This is dependent on the digit 'r', which is the units digit of the number 8pqr.

Let's analyze the given statements individually: 

1. Statement 1: The product of p and q is 12.
   • Possible pairs satisfying this condition are (p, q) = (2, 6), (3, 4), (4, 3), and (6, 2).
   • This statement gives us combinations of p and q but no information about r, so Statement 1 alone is not sufficient to determine the unit's digit of 8pqr.
2. Statement 2: The product of q and r is 24 and r is greater than 4.
   • Possible pairs for (q, r) given the condition are (q, r) = (3, 8), (4, 6), (6, 4).
   • From these, only the pair (q, r) = (3, 8) satisfies the condition r > 4. So r = 8.
   • This statement alone allows us to determine r = 8, which is the unit digit of 8pqr.

Although Statement 2 alone determines r, the context of the problem (Data Sufficiency) means we need extra verification due to possible ambiguous conditions. Let's consider both statements together:

• From Statement 1, possible values of (p, q) are (2, 6), (3, 4), (4, 3), (6, 2).
• From Statement 2, (q, r) pair is (3, 8).
• The only combination where both conditions meet is (p, q, r) = (4, 3, 8), satisfying both q = 3 and r = 8.

This confirms that the unit's digit r = 8.

The pattern for powers of 8 gives the repeating cycle of units digit as: 8, 4, 2, 6. Therefore, \(8^{64}\) will have the same units digit as \(8^{0}\) or \(8^4\), which is 6.

Thus, both statements together are necessary to find the unit digit of \((8pqr)^{64}\).

Conclusion: The correct answer is: Both the statements together are needed to answer the question.