Residue

Number Theory Level 3

Find the remainder when 6 85 + 8 85 6^{85}+8^{85} is divided by 49 49 .


The answer is 14.

Vincent Lopez by Vincent Lopez , 25, Philippines

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Hongqi Wang
Jan 27, 2021

6 85 + 8 85 = ( 7 − 1 ) 85 + ( 7 + 1 ) 85 = 49 m + ( 85 1 ) × 7 − 1 + 49 n + ( 85 1 ) × 7 + 1 ≡ 2 × ( 84 + 1 ) × 7 ≡ 14 ( m o d 49 ) \begin{aligned} 6^{85} + 8^{85} = (7-1)^{85} + (7+1)^{85} = 49m + \binom{85}{1} \times 7 -1 + 49n + \binom{85}{1} \times 7 + 1 \equiv 2 \times (84+1) \times 7 \equiv 14 \pmod {49} \end{aligned}

1 Helpful 1 Interesting 2 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...