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The value of $3^{51} \text{ mod } 5$ is _____

### 4 Comments

@the_pycho, I don’t know why you asked to study equivalence classes for such simple thing. It is basically could be verified simply. -3/5 + 5*a is the equation after dividing it by 5 and a is obviously greater than 1. so we can simply say do this -3/5 + 5 + 5*(a-1), and now simplify to see what is remainder.

Watch these two videos next time these problems will look simple

@KUSHAGRA गुप्ता How did you took 51’s binary form and took it to 3’s power? can you explain why that works?

## 15 Answers

Answer is 2

we will see why

3mod 5 ( $3 \equiv -2 mod 5$)

The remainders are taking an order when divided with 5.

Consider

(3 power 1) mod 5 =  2

(3 power 2) mod 5  = 4

(3 power 3) mod 5  = 2

This will continue for all other powers

Therefore for even number power answer will be 4 and for odd number power answer will be 2

Hence

(3 power 51) mod 5 = 2

by

Answer : 2

cycle of 3 where unit place repeat is 4 . and unit digit of last cycle is 1 .

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

now repeat

3^5 = 3

so, (3^4)^12 * 3^3 mod(5) = 1*3^3 mod(5) = 27 mod(5) = 2

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