Can you find the remainder?

15 9 6 7 77 m o d 17 = ? \Large 159^{67^{77}} \bmod {17} = \ ?

6 5 1 12

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1 solution

. .
Feb 17, 2021

15 9 67 77 m o d 17 = 12 { 159 ^ { 67 } } ^ { 77 } \mod 17 = \boxed { 12 } because 159 159 end up with the number 9 9 , and the last digit when we power 9 is always 1 , 9 1, 9 , but the 2 exponents are all odd, so 9 9 is the last digit.

And 9 m o d 17 = ? ? , 19 m o d 17 = 2 , 29 m o d 17 = 12 , 39 m o d 17 = 5 , 49 m o d 17 = 15 , 59 m o d 17 = 8 , 69 m o d 17 = 1 , e t c . 9 \mod 17 = ??, 19 \mod 17 = 2, 29 \mod 17 = 12, 39 \mod 17 = 5, 49 \mod 17 = 15, 59 \mod 17 = 8, 69 \mod 17 = 1, etc. . So the answer is 12 \boxed { 12 } .

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