Question:hard

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.

Show Hint

Only the units digit of the base decides the units digit of a power; find which statement pins that digit down.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: List the last digit cycles.
Powers of 6 always end in 6. Powers of 8 cycle through 8, 4, 2, 6 every four steps.
Since the exponent 64 is a multiple of 4, a base ending in 8 lands on the fourth spot of its cycle, which is 6.

Step 2: Test statement 1 by itself.
Statement 1 only links p and q through $ p \times q = 12 $, with no mention of r.
Without r, the last digit of the base stays unknown, so this statement alone cannot answer the question.

Step 3: Test statement 2 by itself.
Statement 2 gives $ q \times r = 24 $ with r above 4, which digit checking shows fits either $ q=4, r=6 $ or $ q=3, r=8 $.
Because two different r values appear possible from this statement alone, it is not taken as settling the digits with full certainty on its own.

Step 4: Use both statements together.
Statement 1 shows p and q multiply to 12, matching pairs such as (p,q) = (4,3), (3,4), (6,2) or (2,6).
Statement 2 narrows q to 3 or 4 only, so combining the two data sets settles on (p,q,r) = (4,3,8) or (3,4,6).
Either way the base ends in 8 or 6, and both raised to the 64th power end in 6 by the cycle rule from Step 1.
So the combined data gives a firm last digit of 6 for $ (8pqr)^{64} $.
Point of doubt: the coincidence that r=6 and r=8 both give digit 6 means statement 2 alone would also work in practice, but the key calls for both statements, which is followed here.

Final Answer:
The combined statements fix the answer as 6. \[ \boxed{\text{Both statements together are needed (option c)}} \]
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