The Step Cypher

Number Theory Level 2

What is the last digit of the number ${ 23 }^ { \left( { 6 }^{ 1994 } \right) }?$

2 solutions

Hanif Robbani
Aug 1, 2014

Considering only the last digit, $23^6$ equals to $3^6$ , their last digit is 9.

We can observe that last digit of $9^n$ has a pattern :

... when $n$ is divisible by 2, the last digit is 1. Since 1994 is divisible by 2, then last digit of $(23^6)^{1994}$ is $\boxed{1}$

Sunil Pradhan
Jul 23, 2014

23^6^1994 consider only unit places = 2^6^4 = 2^6^1996

To find unit digit Index ÷ 4 find remainder 1996/4 remainder is 0

unit digit = 3^4 = 81 i.e. 1

In the first line, I think 1996 at the end is a typo? It should be 1994, and this persists to the second line.

Can you add more explanation to the second line?

Calvin Lin Staff - 6 years, 10 months ago