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Difficulty: MediumUnit Digit and Cyclicity

What is the unit digit of the expression E=138124+26493−17754E = 138^{124} + 264^{93} - 177^{54}?

  1. 1Answer
  2. B
    6
  3. C
    9
  4. D
    3

Answer

The unit digit of the expression is 1.
The correct answer is derived by determining the cyclicity of each base's unit digit: 8 has cyclicity 4 (with remainder 0 yielding unit digit 6), 4 has cyclicity 2 (odd exponent yielding unit digit 4), and 7 has cyclicity 4 (remainder 2 yielding unit digit 9). Combining these yields (6 + 4) - 9 = 1.

Step-by-Step Solution

1
Find the unit digit of 138124138^{124}
Unit digit is 6
The unit digit of the base is 8. The cyclicity of 8 is 4 (8, 4, 2, 6). Dividing the exponent 124 by 4 gives a remainder of 0. A remainder of 0 corresponds to the 4th power in the cycle (84=40968^4 = 4096), so the unit digit is 6.
2
Find the unit digit of 26493264^{93}
Unit digit is 4
The unit digit of the base is 4. The cyclicity of 4 is 2 (41=4,42=64^1 = 4, 4^2 = 6). Since the exponent 93 is odd, the unit digit is 4.
3
Find the unit digit of 17754177^{54}
Unit digit is 9
The unit digit of the base is 7. The cyclicity of 7 is 4 (7, 9, 3, 1). Dividing the exponent 54 by 4 gives a remainder of 2 (54=4×13+254 = 4 \times 13 + 2). Thus, the unit digit is 72=497^2 = 49, which ends in 9.
4
Combine the unit digits according to the expression
Unit digit is 1
Evaluating (6+4)−9=10−9=1(6 + 4) - 9 = 10 - 9 = 1. The unit digit of the entire expression is 1.

Key Concept

Unit Digit Cyclicity and Modular Exponentiation Rules
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Topics

Unit Digit and Cyclicity

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