Q1

For each question, you should clearly show steps about how you get the answer.

a) Prove or disprove the statement. For all integers x, y and z, if x|y and x|z, then x|^{(2 2)}

b) Compute the values of n based on div. For which values of n is n div 5 = -2?

c) Calculate the integer n based on div and mod:

i. n div 4 = 7, n mod 4 = 3

ii. n div 6 = 10, n mod 6 = 4

Q2

a) Give the multiplication and addition tables for ₇.

b) Compute the final answer below without using a calculator. Show all your steps.

i. 32⁹ 9

Q3

Compute the value of the following questions. (For each question, you should clearly show steps about how you get the answer.)

a) (72) div 6

b) (155) mod 12

c) Suppose n mod 4 = 2, find all possible values of n.

d) [(89 mod 13) + (47 mod 13)] mod 13

Q4

Give the prime factorization for each number in non-decreasing order (For each question, you should clearly show steps about how you get the answer.)

a) 120

b) 168

c) 315

d) 292

Q5

Find the greatest common divisor (gcd) and least common multiple (lcm). (For each question, you should clearly show steps about how you get the answer.)

a) (56, 98)

b) (84, 126)

c) (282, 352)