Need Math 324 homework solution Alberta University

Question - Math 324 Assignment 3 (due October 1)

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1. (3.5, #14)(5) How many zeros are there at the end of 1000! in decimal notation?

2. (3.5, #56)(6) Prove that there are infinitely many primes of the form 6k + 5 where k is
a positive integer (without quoting Theorem 3.3). Hint: Adapt the proof of Theorem 3.17.

3. (3.5, #70)(5) Show that every prime p divides the binomial coefficient for all
integers k with 0 < k < p.
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4. (4.1, #32)(5) Give a complete system of residues modulo 13 consisting entirely of odd
integers.

5. (4.1, #36)(5) Find the least positive residue m ...Read More od 47 of 232.

6. (7) Find all incongruent solutions mod 209 of x2 − 3x + 1 ≡ 0 mod 209.

7. (2 + 3 + 2) (Continuing Problem 2.7) Define π ∈ A to be prime if π is not a unit and
the only way to get π = αβ with α, β in A is with one of α, β a unit. Primes π, π′ are
associates if uπ = π′ for some unit u.
a) Find all units of A.
b) Find all primes π with N(π) ≤ 10, up to associates.
c) Show that every non-unit α ∈ A has a prime factor π. Hint: Consider the subset
{N(α): α non-unit of A} of N. ...Read Less

Solution Preview - iggest power of 10 that divides 1000! Using Exercise 12 of section 3.5 and 10 = 2⋅5, this is the smaller of ∑ r ≥ 1 [1000/2r] and ∑ r ≥ 1 [1000/5r]. The first sum is bigger (by direct calculation or a general argument) than the second, which is [1000/5] + [1000/25] + [1000/125] + [1000/625] + 0 = 200 + 40 + 8 + 1 = 249. 2. Suppose t

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