3 - Playing With Remainders

Number Theory Level 2

What is the remainder when 16 32 {16}^{ 32 } is divided by 11.


The answer is 3.

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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1 solution

Maggie Miller
Aug 7, 2015

Note 5 5 = ( 25 ) 2 ⋅ 5 ≡ 3 2 ⋅ 5 = 45 ≡ 1 ( m o d 11 ) 5^5=(25)^2\cdot 5\equiv 3^2\cdot 5=45\equiv 1\pmod{11} . Then

1 6 32 ≡ 5 32 = ( 5 5 ) 6 5 2 ≡ 1 6 ⋅ 25 ≡ 3 ( m o d 11 ) 16^{32}\equiv5^{32}=(5^5)^65^2\equiv 1^6\cdot 25\equiv \boxed{3}\pmod{11} .

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