Question

Show that if a number ends with the digit 1, then all power of this number end with 1. Please prove the correctness of your analysis.

Answer 1

To compute the power of any number, It must be multiplied by itself multiple times..
So, If a number which has a LSB digit as 1, is multiplied by itself, then the result always has the Least significant byte as 1.. Like:

11 * 11 = 121
21 * 21 = 441, And so on..

Hence if N ends on 1,
Then, N * N will end on 1..
Similarly, N * N * N = (10x + 1) * N where x can be some integer.. We represented N*N = 10x + 1, Because it resulted in a number with last digit as 1.

Hence (10x + 1) * N will also result in LSB as 1, Because we already proved that any 2 numbers having LSB as 1, result into LSB as 1 in the product.

Similarly, Any number with LSB as 1, multiplied multiple times will result in LSB 1.