Question

Whats the remainder when 2018 to the power 2018 is divided by 20???

Answer 1

Answer:

4

Step-by-step explanation:

Answer 2

Answer:4

This requires understanding of modular arithmetic and certain subtraction and exponent rules.

(Whenever I use an equal sign to express mods, the equal sign represents the congruence sign)

Realize that 2018=18(mod 20)=

(-2)(mod 20)

Exponent rule for mod:

Given that a=b (mod n)

a^k= (b^k) (mod n)

Therefore,

2018^2018= [(-2)^2018] (mod 20)

= [(2)^2018] (mod 20)

Let's look at the last digit when we raise 2 to some power

2^1 has last digit of 2

2^2 has last digit of 4

2^3 has last digit of 8

2^4 has last digit of 6

This pattern repeats

To figure out what 2^(2018) end in, divide 2018 by 4, raise 2 to the remainder and take the last digit of that number.

2018/4 has remainder of 2.

2^2=4 which ends in 4.

So 2^(2018)=............4 (this means number ends in 4)

(2018)^(2018)=(..........4) (mod 20)

We can keep subtracting 20 until we reach a number less than 20.

Realize that subtracting a number by 20 will not affect the last digit of the number as long as the number is greater than 20.

If we subtract 20 enough times we will reach a number that is bigger than 20 but still ends in 4.

In other words,

(2018^2018)=(20a + 4) (mod 20)

We subtract 20 "a" more times and we end with:

(2018^2018)=4 (mod 20)

The number before the mod represents remainder.

Hence, remainder is 4