What is the unit digit of the expression 8^{20} + 7^{13}? — State PSC Exam

What is the unit digit of the expression $8^{20} + 7^{13}$ ?

The unit digit of the given expression is 3.

The unit digit of

8^{20}

is 6 because 20 is a multiple of 4, matching the 4th term in the cyclicity cycle (8, 4, 2, 6). The unit digit of

7^{13}

is 7 because 13 leaves a remainder of 1 when divided by 4, matching the 1st term in the cyclicity cycle (7, 9, 3, 1). Adding these unit digits gives

6 + 7 = 13

, whose unit digit is 3.

Find the unit digit of

8^{20}

Unit digit is 6

The cyclicity of numbers ending in 8 is 4 (pattern: 8, 4, 2, 6). Since 20 is divisible by 4 (

20 \pmod 4 = 0

), the unit digit corresponds to the 4th power in the cycle, which is 6.

Find the unit digit of

7^{13}

Unit digit is 7

The cyclicity of numbers ending in 7 is 4 (pattern: 7, 9, 3, 1). Dividing 13 by 4 gives a remainder of 1 (

13 \pmod 4 = 1

), so the unit digit corresponds to

7^1 = 7

Add the unit digits of both terms

Sum of unit digits is

6 + 7 = 13

. Taking the unit digit of 13 gives 3.

Cyclicity of Unit Digits

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