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**Question - **Math 324 Assignment 3 (due October 1)
Please remember to staple your assignment and to write the names of other students you
may have worked with.
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.
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
k
p
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 mod 47 of 232.
6. (7) F
...Read More

**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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