Need help on maths problem

Aloha1990 SG

A student is exploring numbers of the form 3^n, where n is any positive integer. She makes a table of results shown in Attachnment (a). From the table, she observes that whenever n is a muitple of 4, the last digit of the number produced from 3^2470 should be a 9. Give a brief explanation of the mathmetical reasoning she might have used to make this statenment. As a result,what di you think ger conjecture might have been?

N 3^N last digit
0 1 1
1 3 3
2 9 9
3 27 7
4 81 1
5 243 3
6 729 9
7 2187 7
8 6561 1
9 19683 3

Aloha1990 SG

upz upz upz

frekiwang SG

conjecture:

The last digit of 3^n is

3 if n=4k+1

9 if n=4k+2

7 if n=4k+3

1 if n=4k

for all non-negative integers k.

Proof,

a) You can use MI (if you are taking H2 math in JC), start with P0,P1,P2,P3 then do the induction based on assumption.

b) You can use the properties of mod function, if you are doing math olympiad or University math,

c) You can forget about the proof if you are in primary school.