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Question - Math 324 Assignment 4 (due October 8) Please remember to staple your assignment and to write the names of other students you may have worked with. 1. ( 5)(1.5, #46) Show that the nth Fibonacci number fn is divisible by 4 if, and only if, 6 divides n. Hint: Consider the sequence of least residues of fn mod 4. 2. (5)(3.5, #34) Find all pairs of positive integers a, b so that (a, b) = 18 and [a, b] = 504. 3. (6) Determine all solutions x mod 147 of 119x ≡ 98 mod 147. 4. ( 5) Let p be an odd prime and k ≥ 1. What are the solutions mod pk of the congruence x2 ≡ 1 mod pk ? Prove this. 5. ( 3 + 3)(Continuing Problem 3.7) a) Show that every non-unit of A can be written as a product of primes of A. b) Is prime fa ...Read More

Solution Preview - starts as 0, 1, 1, 2, 3, 1, 0, 1 for 0 ≤ n ≤ 7, which is long enough to observe that f n + 6 ≡ f n mod 4 for n = 0, 1, hence for all n ≥ 0 by induction on n: for n ≥ 2, we have f n + 6 = f n − 1 + 6 + f n − 2 + 6 ≡ f n − 1 + f n − 2 = f n mod 4. For 0 ≤ r < 6, another induction, started by the one above, gives f 6q + r ≡

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