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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 â‰¡ 98 mod 147.
4. ( 5) Let p be an odd prime and k â‰¥ 1. What are the solutions mod pk of the congruence
x2 â‰¡ 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)
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**Solution Preview - **starts as 0, 1, 1, 2, 3, 1, 0, 1 for 0 â‰¤ n â‰¤ 7, which is long enough
to observe that f n + 6 â‰¡ f n mod 4 for n = 0, 1, hence for all n â‰¥ 0 by induction on n:
for n â‰¥ 2, we have f n + 6 = f n âˆ’ 1 + 6 + f n âˆ’ 2 + 6 â‰¡ f n âˆ’ 1 + f n âˆ’ 2 = f n mod 4. For 0 â‰¤ r < 6,
another induction, started by the one above, gives f 6q + r â‰¡

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