Multiple choice general knowledge

For how many integers n, with 10<=n<=100 is n*n+n-90 divisible by 17?

  1. 7

  2. 8

  3. 9

  4. 10

  5. 12
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D Correct answer
Explanation

Need n² + n - 90 ≡ 0 (mod 17), which means (2n+1)² ≡ 361 ≡ 1 (mod 17), so 2n+1 ≡ ±1 (mod 17). This gives n ≡ 0 or 12 (mod 17). In [10,100]: n = 12, 29, 46, 63, 80, 97 (for n ≡ 12) and n = 17, 34, 51, 68, 85 (for n ≡ 0). Total = 10 values.

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