Question

Consider the powers of 7. What is the last digit of 7 to the 1000?

Answer 1

Answer:

Let’s check if there is any rule in powers of 7:


`image`

We can see every 4 index of power of 7 ending is the same (for example 7^6 ending is 9, and 7^10 is also 9), so now we can count:

(„*” stand for digits before last numbers)

7^4 = ***1
7^8 = ***1
7^16 = ***1
7^40 = ***1
7^160 = ***1
7^960 = ***1
7^960 + 7^40 = ***1

So the last digit of 7^1000 will be one.

Hope this helps!

Answer 2

Hi

1

Explanation:

`\begin{array}exponent&last\ digit\\0&1\\1&7\\2&9\\3&3\\4&1\\...\\\end {array}\\\\\text{The last digit of } 7^n\ is\\1\ if\ n\in4\mathbb{N}\\7\ if\ n\in4\mathbb{N}+1\\9\ if\ n\in4\mathbb{N}+2\\3\ if\ n\in4\mathbb{N}+3\\\\1000=250*4+0 \in 4\mathbb{N}\\\\last\ digit(7^(1000))=1\\`