What is the unit digit of the expression 2^{52} + 5^{17}? — State PSC Exam

What is the unit digit of the expression $2^{52} + 5^{17}$ ?

The unit digit of the expression

2^{52} + 5^{17}

is 1.

For

2^{52}

, dividing 52 by 4 leaves a remainder of 0. Since the cyclicity of 2 is 4, a remainder of 0 means using

2^4 = 16

, which has a unit digit of 6. For

5^{17}

, any power of 5 ends in 5. The total unit digit is the unit digit of

6 + 5 = 11

, which is 1.

Find the unit digit of

2^{52}

using cyclicity.

The cyclicity of 2 is 4 (pattern: 2, 4, 8, 6). Dividing the exponent 52 by 4 yields

52 = 4 \times 13 + 0

. A remainder of 0 corresponds to the 4th power in the cycle.

2^4 = 16

, so the unit digit of

2^{52}

is 6.

When the exponent is exactly divisible by the cyclicity length, the unit digit corresponds to the power equal to the cyclicity length.

Find the unit digit of

5^{17}

The cyclicity of 5 is 1, meaning any positive integer power of 5 ends in 5. Thus, the unit digit of

5^{17}

is 5.

Base numbers ending in 5 always yield a unit digit of 5 for any positive integer exponent.

Add the individual unit digits to get the final unit digit.

Unit digit of

(2^{52} + 5^{17}) = \text{Unit digit of }(6 + 5) = \text{Unit digit of }11 = 1

The unit digit of a sum is equal to the unit digit of the sum of the individual unit digits.

Unit Digit and Cyclicity

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