Remainders Revisited #1

Number Theory Level 3

8 5 1 9 51 m o d 51 = ? \Large 85^{19^ {51}} \bmod 51 = \, ?


The answer is 34.

Ravneet Singh by Ravneet Singh , 30, India

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1 solution

Revanth Gumpu
Aug 4, 2015

Note that 34^2 = 34 mod 51. This fact will help us. 85^19^51 is equal to 85^969. Using the fact previously stated, we get: 85^969 = 34^969 = 34^484 * 34 = 34^242 * 34 = 34^122 = 34^61 = 34 ^ 30 * 34 = 34^16 = 34^8 = 34^4 = 34^2 = 34. Therefore 85^19^51 = 34 mod 51.

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Moderator note:

I disagree with "85^19^51 is equal to 85^969".

Read Rules of Exponents to recall how the tower of exponents is calculated.

Calvin Lin Staff - 5 years, 10 months ago

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My bad. How would I solve this then?

Revanth Gumpu - 5 years, 10 months ago

A better question would be how did I still get the right answer?

Revanth Gumpu - 5 years, 10 months ago

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You got the right answer, because 3 4 n ≡ 34 ( m o d 51 ) 34^ n \equiv 34 \pmod{51} for all positive integers n n . Hence, it didn't matter which power you raised it to.

Calvin Lin Staff - 5 years, 10 months ago

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