Number theory

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Group A: (Please answer five questions only.) (10 points each: 50 total]
1. Prove that if gcd(a, b) = 1 and ged(a, c) = 1, then god(a, bc) = 1.
(You may use only the material from chapters 2-6 of our workbook and textbook)
2. Prove that if a positive integer n has k distinct odd prime factors, then 2*|®(n).
(You may use only the material from chapters 2-11 of our workbook and textbook)
3. (a) Given that the congruence 71734250 = 1660565 (mod 1734251) is true, can you make
any conclusion about the primality (or not) of 1734251?
(b) Given that the congruence 252632 = 1 (mod 52633) is true, can you make any con-
clusion about the primality (or not) of 64027?
(You conclusions can be based only on the given information.)
4. Let p be a prime number (P + 2 and p + 5) and let A be some given number. Suppose
that p divides the number A² – 5. Show that p must be congruent to either 1 or 4
modulo 5.
5. A farmer bought 21 birds for a total of $163. The birds are of three types: chickens,
ducks, and geese. The chickens cost $3 each, the ducks cost $5 each, and the geese
cost $13 each. If the farmer bought at least 4 chickens, and at least as many geese as
chickens, then how many of each type of bird did the farmer buy?
Note: The correct answer alone will receive only 6 points credit. To receive full credit,
you must prove that your solution is the only possible solution.
(Hint: Solve this problem using the techniques for solving linear Diophantine equations presented in
Chapter 6. Because the total number of birds bought is know, one can reduce this problem to a linear
Diophantine equation in two variables.)
6.
a
6
3
8
1 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18
I(a) 18 1 13 2 16 14
17 12 15 5 7 11 4 10 9
Use the table of indices modulo 19 for the base 2 (given above) to solve the following
congruence:
152 10
= 3
(mod 19
7. Determine if the congruence x2 = -15 (mod 227) have any solutions. (227 is a prime.)