Math 40520 Number Theory, Evens Worksheet
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1
2HOMEWORK 3, MATH 40520, NUMBER THEORY, EVENS.
Due Wednesday, September 14
INSTRUCTIONS: Do 9 of these 13 problems.
1. Let p be a prime of the form 3k + 1 with k ∈ Z
. Prove that p is of the form 6k + 1 with
>0
k ∈ Z
.
>0
2. Problem 17 of 1.3, p. 29.
2
2
2
3. For a, b ∈ Z, not both 0, prove that (a
) = ((a, b))
.
, b
4. Prove that there are infinitely many primes of the form 4n + 3, and that there are infinitely
many primes of the form 6n + 5.
5. Prove that n divides (n
1)! for every composite number n > 4. If p is a prime, does p divide
(p
1)!?
4
6. Show that n
+ 4 is a composite number for every n > 1.
7. Problem 43, of 1.3, p. 33.
8. Let a = 38808 and let b = 1887600.
(a) Use the Euclidean algorithm to compute (a, b).
(b) Find the prime factorizations of a and b.
(c) Use your answer in part (b) to find the prime factorizations of (a, b) and [a, b].
9. List all numbers from 1 to 150 that are congruent to 5 mod 23.
10. For each of the following integers n, find a reduced residue system modulo n:
(a) n = 7
(b) n = 14
(c) n = 20.
11. Let n be an integer. Prove the following assertions:
12
(a) n
1 is divisible by 7
16
(b) n
1 is divisible by 17
80
(c) n
1 is divisible by 17.
12. Let a
be the decimal representation of a number m with n + 1 digits. Prove
a
. . . a
a
n
n 1
1
0
that 11 divides m if and only if 11 divides
n
i
( 1)
a
.
i
i=0
13. For a number m with 4 digits in the form a
, give a criterion for when 7 divides m.
a
a
a
3
2
1
0
How much time does it take to apply compared to just punching the numbers into a calculator?
(hint: for a number m with 2 digits, m = a
, then m is divisible by 7 if and only if 7 divides
a
1
0
+ 3a
).
a
0
1
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