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do you know how to do 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.)