Adaptable

Number Theory Level 2

Find the remainder when 1 5 23 + 2 3 23 15^{23} + 23^{23} is divided by 19.


The answer is 0.

Abhin Rfad by abhin rfad , 23, India

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3 solutions

Alan Yan
Dec 30, 2015

15 ≡ − 4 (mod 19) 15 \equiv -4 \text{ (mod 19) } 23 ≡ 4 (mod 19) 23 \equiv 4 \text{ (mod 19) } Therefore the desired expression is ( − 4 ) 23 + ( 4 ) 23 ≡ 0 (-4)^{23} + (4)^{23} \equiv \boxed{0} .

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Drex Beckman
Jan 2, 2016

I'm not sure if this is correct, just an observation. But I'm throwing this out there: x n + y n ≡ 0 ( m o d x + y ) x^{n}+y^{n}\equiv0\hspace{2mm}(mod\hspace{1mm}x+y) So, 1 5 23 + 2 3 23 ≡ 0 ( m o d 38 ) 15^{23}+23^{23}\equiv0\hspace{2mm}(mod\hspace{1mm}38) Since it is divisible by 38, and 38=19*2, it will likewise be 0(mod 19).

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Ramiel To-ong
Jan 9, 2016

nice solution

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