Question

What are the last three digits in the integer equal to 5^2020?

Answer 1

Answer:

100

Step-by-step explanation:

5^2020=10,100

:the last three digits are 100

Answer 2

Answer:

As can be seen, after the 3rd power, the last three digits are 125 and 625 alternatively. 125 for odd powers, 625 for even powers. As 2020 is even, last three digits of 52020 should be 625.

Step-by-step explanation:

Let’s prove it.

52n≡625(mod1000)∀n>1.

Induction is not a bad idea.

For n=2 we have

52⋅2=54=625≡625(mod1000).

Ok.

Now let’s assume that the relation is true for n=k

52k≡625(mod1000)

and let’s prove it for n=k+1 :

52(k+1)=52k⋅52≡625⋅25=15625≡625(mod1000).

Ok.

Therefore, it is true for all n>1 .