Question

Decide if the following statements are true or false. Justify your answer.

a) If p is prime, then 22p - 1 is prime. True False
b) Let a, b, c, and n be positive integers. If ab = ac mod n, then b = c mod n. True False
c) Let a, b, and c be positive integers. If a | c and b | c, then ab | c. True False

Answer 1

a False

b False

c. True

Explanation:

a) False, because if for example p = 3, then 22p - 1 = 66 - 1 = 65 which is not prime.

b) Lets look at an example

ab = ac mod n

Let a = 3, b = 4 and c = 9:

12 = 27 mod 15 ( as 12 is the remainder when 27 is divided by 15)

Divide ab and ac by a ( = 3) we have

4 = 9 mod 15

bt 9 mod 15 is 9

so FALSE.

c)

a|c and b|c means a divides exactly into c and b divides exactly into c.

So we can write ak = c and bl = c where k and l are 2 integers ( not = 0).

Therefore ak = bl

so a^2b^2kl = c^2 which is divisible by ab.

True