Remains when Divided

Number Theory Level 3

Find the remainder of the following :

6 83 + 8 83 6^{83}+ 8^{83} when divided by 49 .


The answer is 35.

Anish Harsha by Anish Harsha , 20, India

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1 solution

Maggie Miller
Aug 28, 2015

The totient ϕ ( 49 ) = 49 ( 1 − 1 7 ) = 42 \phi(49)=49\left(1-\frac{1}{7}\right)=42 . Then by Euler's totient theorem, 6 42 ≡ 8 42 ≡ 1 ( m o d 49 ) 6^{42}\equiv 8^{42}\equiv 1\pmod{49} . Therefore, 6 83 + 8 83 ≡ ( 6 42 ) 2 6 − 1 + ( 8 42 ) 2 8 − 1 ≡ 6 − 1 + 8 − 1 ( m o d 49 ) 6^{83}+8^{83}\equiv (6^{42})^26^{-1}+(8^{42})^28^{-1}\equiv6^{-1}+8^{-1}\pmod{49} .

Note 49 − 6 ⋅ 8 = 1 49-6\cdot 8=1 , so modulo 49 49 , 6 − 1 = − 8 6^{-1}=-8 and 8 − 1 = − 6 8^{-1}=-6 . Thus, 6 83 + 8 83 ≡ − 8 − 6 = − 14 ≡ 35 ( m o d 49 ) 6^{83}+8^{83}\equiv-8-6=-14\equiv \boxed{35}\pmod{49} .

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