Question

What is the remainder of 4^31 when divided by 10

Answer 1

The remainder of 4^31 when divided by 10 is 4, determined by examining the repeating last-digit pattern of powers of 4.

Step-by-step explanation:

To find the remainder of 4^31 when divided by 10, we can use the concept of powers and properties of numbers related to modulo arithmetic. Specifically, we look at the last digit pattern that emerges when calculating powers of 4, since the remainder when dividing by 10 depends solely on the final digit of the number.

The powers of 4 have a cycle: 4^1 ends with 4, 4^2 ends with 6, 4^3 ends with 4 again, and so on. This cycle repeats every two powers, so we only need to consider the power of 4 modulo 2 to determine the last digit of 4^31. Since 31 is an odd number, we know that 4^31 will end with a 4. Hence, the remainder of 4^31 divided by 10 is 4.

Answer 2

We start with


`4^1 \equiv 4 \ (\text{mod} \ 10)`

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Square both sides


`4^1 \equiv 4 \ (\text{mod} \ 10)`


`(4^1)^2 \equiv 4^2 \ (\text{mod} \ 10)`


`4^2 \equiv 16 \ (\text{mod} \ 10)`


`4^2 \equiv 6 \ (\text{mod} \ 10)`

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Square both sides again


`4^2 \equiv 6 \ (\text{mod} \ 10)`


`(4^2)^2 \equiv 6^2 \ (\text{mod} \ 10)`


`4^4 \equiv 36 \ (\text{mod} \ 10)`


`4^4 \equiv 6 \ (\text{mod} \ 10)`

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Square both sides again


`4^4 \equiv 6 \ (\text{mod} \ 10)`


`(4^4)^2 \equiv 6^2 \ (\text{mod} \ 10)`


`4^8 \equiv 36 \ (\text{mod} \ 10)`


`4^8 \equiv 6 \ (\text{mod} \ 10)`

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Square both sides again


`4^8 \equiv 6 \ (\text{mod} \ 10)`


`(4^8)^2 \equiv 6^2 \ (\text{mod} \ 10)`


`4^(16) \equiv 36 \ (\text{mod} \ 10)`


`4^(16) \equiv 6 \ (\text{mod} \ 10)`

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So far we have found that,


`4^1 \equiv 4 \ (\text{mod} \ 10)`


`4^2 \equiv 6 \ (\text{mod} \ 10)`


`4^4 \equiv 6 \ (\text{mod} \ 10)`


`4^8 \equiv 6 \ (\text{mod} \ 10)`


`4^(16) \equiv 6 \ (\text{mod} \ 10)`

Multiply out all of the left sides of those equations above. Multiply out the right sides as well to balance things out


`(4^1)*(4^2)*(4^4)*(4^8)*(4^(16)) \equiv 4*6*6*6*6 \ (\text{mod} \ 10)`


`4^(1+2+4+8+16) \equiv 24*36*6 \ (\text{mod} \ 10)`


`4^(31) \equiv 4*6*6 \ (\text{mod} \ 10)`


`4^(31) \equiv 24*6 \ (\text{mod} \ 10)`


`4^(31) \equiv 4*6 \ (\text{mod} \ 10)`


`4^(31) \equiv 24 \ (\text{mod} \ 10)`


`4^(31) \equiv 4 \ (\text{mod} \ 10)`

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The final answer is 4