A number theory problem by Samina Siamwalla

The remainder when 3 12 + 5 12 3^{12}+5^{12} is divided by 13 13 is

4 1 3 2

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2 solutions

Samrit Pramanik
Jun 3, 2018

By Fermat's little theorem, a p a ( m o d p ) a^p\equiv a \pmod{p} , for any a N a \in \mathbb{N} and p p be any prime number

Now, putting a = 3 , 5 a=3,5 and \text{and} p = 13 p=13 we get the followings,

3 13 3 ( m o d 13 ) 3^{13}\equiv 3 \pmod{13} and 5 13 5 ( m o d 13 ) 5^{13}\equiv 5 \pmod{13}

3 12 1 ( m o d 13 ) \Longrightarrow 3^{12}\equiv 1 \pmod{13} and 5 12 1 ( m o d 13 ) 5^{12}\equiv 1 \pmod{13}

and adding these two we get finally

3 12 + 5 12 2 ( m o d 13 ) 3^{12}+5^{12}\equiv 2 \pmod{13}

i.e. \text{i.e.} the remainder is 2 2 when 3 12 + 5 12 3^{12}+5^{12} is divisible by 13 13

Samina Siamwalla
Jun 3, 2018

If 'a' is an integer and 'p' is a prime number and when 'a' is not divisible by 'p', then a^{p-1}=1modp 13 is a prime number which is not divisible by 3 and 5 then 3^{12}=1mod13 and 5^{12}=1mod13 therefore, 3^{12}+5^{12}=2mod13

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