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Difficulty: EasyUnit Digit and Cyclicity

What is the unit digit of 7827^{82}?

Answer: 9

Answer

The unit digit of 7827^{82} is 9.
The unit digits of powers of 7 repeat in a pattern of 4 (7, 9, 3, 1). Dividing the exponent 82 by 4 yields a remainder of 2. The second number in the cyclic pattern is 9, making 9 the unit digit of 7827^{82}.

Step-by-Step Solution

1
Find the cyclicity of the base number 7
The unit digits follow a repeating 4-step sequence: 7, 9, 3, 1.
Powers of 7 cycle every 4 powers because 71=77^1=7, 72=497^2=49, 73=3437^3=343, and 74=24017^4=2401.
2
Divide the exponent by the cyclicity period
82÷4=2082 \div 4 = 20 with a remainder of 2.
The remainder determines the equivalent power position within the 4-step cycle.
3
Determine the unit digit from the remainder
The unit digit of 727^2 is 9.
A remainder of 2 corresponds to 727^2, giving 9.

Key Concept

Unit Digit Cyclicity
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Topics

Unit Digit and Cyclicity

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