Prove or disprove a For any integers a b if 3 ab then 3 a

Prove or disprove: (a) For any integers a, b, if 3 | ab then 3 | a and 3 | b. (b) For any integers a, b, if 3 | ab then 3 | a or 3 | b. (c) For any integers a, b, if 4 | ab then 4 | a or 4 | b. (d) For every prime number p, for any integers a, b, if p | ab then p | a or p | b.

Solution

a)

False

Let, a=2,b=3

ab=6

Hence, 3|ab

But, 3 does not divide a=2

b)

True

3|ab

3 is prime. Hence by definition of a prime number, 3|a or 3|b

c)

False

Let, a=b=2

ab=4

4|ab

But , 4 does not divide a or b

d)

True

p|ab

Then by definiton of prime number, p|a or p|b