Remainder

Number Theory Level 3

What is the remainder when 5 2039 5^{2039} is divided by 41?


The answer is 33.

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Md Zuhair by Md Zuhair , 20, India

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1 solution

Md Zuhair
Sep 29, 2016

We have 5 3 = 2 m o d ( 41 ) 5^3 = 2 mod(41) 5 5 = 25 x 2 m o d 41 5^5 = 25 x 2 mod 41 5 5 = 9 m o d 41 5^5 = 9 mod 41 5 10 = 81 m o d 41 5^{10} = 81 mod 41 5 10 = − 1 m o d 41 5^{10} = -1 mod 41 Now we have 5 2039 = 5 2030 x 5 9 5^{2039} = 5^{2030} x 5^9 now 5 2030 = − 1 2 03 m o d 41 o r 5 2 030 = − 1 m o d 41 5^{2030} = -1^203 mod 41 or 5^2030 = -1 mod 41 .......... (i)

We have 5 3 = 2 m o d ( 41 ) 5^3 =2 mod (41) or 5 9 = 8 m o d ( 41 ) 5^9 = 8 mod (41) .................... (ii)

Now multiplying (i) and (ii) we get , 5 2039 = − 8 m o d ( 41 ) 5^{2039} = -8 mod (41)
5 2039 = 33 m o d ( 41 ) 5^{2039} = 33 mod (41)

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