Question:medium

What is the unit's digit of the expression 77920 + 64165 + 53246?

Updated On: Jun 30, 2026
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The Correct Option is A

Solution and Explanation

The correct answer is option (A):
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Let's determine the unit digit of each term separately and then combine them.

For 77920, the unit digit is determined by the unit digit of the base, which is 7. We need to find the cycle of powers of 7:
71 = 7
72 = 49 (unit digit 9)
73 = 343 (unit digit 3)
74 = 2401 (unit digit 1)
75 = 16807 (unit digit 7)

The cycle is 7, 9, 3, 1, and it repeats every 4 powers. To find the unit digit of 77920, we consider the exponent 920 modulo 4. 920 divided by 4 is 230 with a remainder of 0. When the remainder is 0, we consider the 4th term in the cycle. Therefore, the unit digit of 77920 is 1.

For 64165, the unit digit is determined by the unit digit of the base, which is 4. We need to find the cycle of powers of 4:
41 = 4
42 = 16 (unit digit 6)
43 = 64 (unit digit 4)
44 = 256 (unit digit 6)

The cycle is 4, 6, and it repeats every 2 powers. To find the unit digit of 64165, we consider the exponent 165 modulo 2. 165 divided by 2 is 82 with a remainder of 1. Therefore, the unit digit of 64165 is 4.

For 53246, the unit digit is determined by the unit digit of the base, which is 3. We need to find the cycle of powers of 3:
31 = 3
32 = 9
33 = 27 (unit digit 7)
34 = 81 (unit digit 1)
35 = 243 (unit digit 3)

The cycle is 3, 9, 7, 1, and it repeats every 4 powers. To find the unit digit of 53246, we consider the exponent 246 modulo 4. 246 divided by 4 is 61 with a remainder of 2. Therefore, the unit digit of 53246 is 9.

Now we add the unit digits we found: 1 + 4 + 9 = 14. The unit digit of the sum is 4. The unit digit of 14 is 4. However, after further review, the remainder of 920/4 is 0, so the unit digit of 77920 is 1. The unit digit of 64165 is 4. The remainder of 246/4 is 2, so the unit digit of 53246 is 9. Adding the unit digits: 1 + 4 + 9 = 14. The unit digit of 14 is 4. I made an error in the original posting, the unit digit of the sum should be 4.

The unit digit of the sum should be 1 + 4 + 9 = 14. The unit digit of 14 is 4. My previous statement was wrong. The correct answer should be 4. The initial answer was incorrect.

We have:
Unit digit of 77920 is 1.
Unit digit of 64165 is 4.
Unit digit of 53246 is 9.
Sum of unit digits is 1 + 4 + 9 = 14. The unit digit of the sum is 4.
The provided answer is incorrect.

Final Answer: The final answer is $\boxed{4}$
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