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# Sol3 - Math 480 HOMEWORK solutions#3 W1 Find all integer...

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Math 480 HOMEWORK solutions # 3 W1. Find all integer solutions of the equation 2 x + 3 y = 11 Answer. (1 + 3 t, 3 - 2 t ) , t Z . W2. (a) For which n is it possible to simplify the fraction 39 n +8 65 n +13 ? Solution. The fraction 39 n +8 65 n +13 is reducible if and only if GCD(39 n + 8 , 65 n + 13) > 1. To find the GCD, we employ Euclidean algorithm. 65 n + 13 = (39 n + 8) + (26 n + 5) 39 n + 8 = (26 n + 5) + (13 n + 3) 26 n + 5 = (13 n + 3) + (13 n + 2) 13 n + 3 = (13 n + 2) + 1 Hence, the GCD=1; and the fraction is already reduced. (b) Solve in integers: x 3 + 21 y 2 + 5 = 0. Solution. Consider the equation mod 7. We have x 3 ≡ - 5 2 (mod 7); hence, x 6 4 (mod 7). This is impossible by the Fermat’s Little theorem (which says that either x 0 or x 6 1 (mod 7)). W3. Recall that the sequence of Fibonacci numbers is defined by F 1 = F 2 = 1, F n +1 = F n + F n - 1 if n 2 . Show that if F n is divisible by p for some n 0, then there are infinitely n such that F n is divisible by p . Solution. Fix a prime p such that there exists n 0 with p | F n 0 . Consider now pairs of consecutive Fibonacci numbers ( F 0 , F 1 ) , ( F 2 , F 3 ) , . . . . By the pigeonhole principle, there must be two pairs ( F i , F i +1 ) and ( F j , F j +1 ), i < j , such that the remainders mod p are pairwise the same, that is F j F i (mod p ) and F j +1 F i +1 (mod p ). Let d = j - i , and let a n = F n + d - F n , n 0. We have a n +2 = F n +2+ d - F n +2 = F n +1+ d + F n + d - F n +1 - F n = a n +1 + a n By the choice of d , we also have a i a i +1 0(mod p ) The recurrence relation now implies that a n 0(mod p ), for any n 0. Hence, F n + d F n (mod p ) for any n 0. By assumption, there exists n 0 such that p | F n 0 . Hence, p | F n 0 + dk for any k 0.

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W4. (a) Show that 17 121 + 5 1331 is divisible by 22. Solution. We prove that for any a, b of the same parity, p > 2 a prime, and i, j non-negative integers, a p i + b p j is divisible by 2 p if a + b is divisible by p .
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Sol3 - Math 480 HOMEWORK solutions#3 W1 Find all integer...

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