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Number Theory
Level
2
Find the remainder when the above expression is divided by 97?
The answer is 0.
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1 solution
This expression can be written as:
1 9 6 + 2 9 6 + 3 9 6 + 4 9 6 + . . . . . . . . . . . . + 9 6 9 6 + 1
And If p is prime then(can be proved by fermat theorem)
1 p − 1 + 2 p − 1 + 3 p − 1 + . . . . . . . . . . . + ( p − 1 ) p − 1 + 1 ≡ 0 ( m o d p )
Since 97 is prime we can place 97 as the value of p.Therefore
1 9 7 − 1 + 2 9 7 − 1 + 3 9 7 − 1 + 4 9 7 − 1 + . . . . . . . . . . . . + 9 6 9 7 − 1 + 1 ≡ 0 ( m o d 9 7 )
Remainder= 0