Show that the units digit in the square of any integer n mus

Show that the units digit in the square of any integer n must be 0, 1, 4, 5, 6, or 9. Follow the instructions of the hint of the previous exercise with m = 3, then m = 4, then m = 5. In each case, what are the possible remainders of n2 on division by m?

Solution

11.

We are basically looking at squares of integers modulo 10

So we can just look at square of integers from 0 to 9 modulo 10

0^2=0

1^2=1

2^2=4

3^2=9

4^2=16=6

5^2=25=5

6^2=36=6

7^2=49=9

8^2=64=4

9^2=81=1

Hence proved

12.

For 3 we need to consider:0,1,2

0^2=0

1^2=1

2^2=4=1

So,0 and1 are the remainders

For 4 we need to conside:0,1,2,3

0^2=0

1^2=1

2^2=4=0
3^2=9=1

So, remainders are:0 and 1

Now for 5 we need to consider:0,1,2,3,4

0^2=0

1^2=1

2^2=4

3^2=9=4

4^2=16=1

So, 0 ,1 and 4