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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

Best Answer 

 #1
avatar+92753 
+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
avatar+92753 
+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

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