What is the last digit of 7^399 :O

Saw this on one Maths Olympiad paper(That's a kind of mathematic competition)

MaxWong May 12, 2016

#1**+21 **

Hi Max

What is the last digit of 7^399

7^1 ends in 7

7^2 ends in 9

7^3 ends in the last digit of 9*7 which is 3

7^4 ends in the last digit of 3*7 which is 1

7^5 ends in the last digit of 1*7 which is 7

so now we have the pattern.

7,9,3,1,7,9,3, etc

there are four numbers in the pattern

7^(4n+1)=7

7^(4n+2)=9

7^(4n+3)=3

7^(4n) = 1

where n is a integer greater or equal to 0.

399 = 400-1 = -1 mod 4 which is the same as 3 mod 4

or if you would rather

399= 4*99+3 see the remainder is 3 so 399=3 mod4

7^(399) = 7^(99*4+3) So the last digit will be **3**

Melody May 12, 2016

#1**+21 **

Best Answer

Hi Max

What is the last digit of 7^399

7^1 ends in 7

7^2 ends in 9

7^3 ends in the last digit of 9*7 which is 3

7^4 ends in the last digit of 3*7 which is 1

7^5 ends in the last digit of 1*7 which is 7

so now we have the pattern.

7,9,3,1,7,9,3, etc

there are four numbers in the pattern

7^(4n+1)=7

7^(4n+2)=9

7^(4n+3)=3

7^(4n) = 1

where n is a integer greater or equal to 0.

399 = 400-1 = -1 mod 4 which is the same as 3 mod 4

or if you would rather

399= 4*99+3 see the remainder is 3 so 399=3 mod4

7^(399) = 7^(99*4+3) So the last digit will be **3**

Melody May 12, 2016