Unformatted text preview: f (18) is divided by 19 2 . 3 (5 points) . Determine the number of solutions to the congruence x 2 + x +1 ≡ mod 7 11 . 4 (6 points) . Find an integer n ≥ 1 so that a 3 n ≡ a mod 85 for all integers a that are divisible neither by 5 nor by 17. 5 (6 points) . Find the number of solutions mod 120 to the system of congruences x ≡ ± 2 mod 4 3 mod 5 4 mod 6 . 6 (7 points) . If m = 15709, we have 2 ( m1) / 2 ≡ 1 mod m and 2 ( m1) / 4 ≡ 2048 mod m . With the aid of these congruences, one can ﬁnd quite easily a positive divisor of m that is neither 1 nor m . Explain concisely: how to ﬁnd such a divisor, and why your method works....
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 Spring '08
 ISBER
 Division, Natural number, Prime number, K. A. Ribet, Professor K. A. Ribet

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