R^r

Number Theory Level 2

Unit digit of 17^17=?

9 3 1 7

Masbah-ul Alam by Masbah-ul Alam , 29, Bangladesh

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2 solutions 

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17**4 --> 1 ( mod 10 )

17**17 = 17**( 4*4 + 1 )

(17**4)**4 --> 1 (mod 10)

17**17 --> 7 (mod 10)

2 Helpful 0 Interesting 0 Brilliant 0 Confused
John Lesteя Tan
Aug 14, 2015
  • 17 ^ 17 mod 10 -7^17 mod 10
  • as you can see 7 has a recurring pattern -7^1 = 7 mod 10
  • 7^2 = 9 mod 10
  • 7^3 = 3 mod 10
  • 7^4 = 1 mod 10
  • so every 4 it becomes 1 mod 10
  • (7^4)^4 * 7 = 1 * 7 mod 10
  • 7 mod 10.
1 Helpful 0 Interesting 0 Brilliant 0 Confused

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