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. For any integers a, b, if 3 | ab then 3 | a or 3 | b. For any integers a, b. if 4 ab then 4| a or 4 | b. (Extra Credit) For every prime number p, for any integers a, b, if p | ab then p | a or p | b.

Solution

a)

False

Let, a=1,b=3

ab=3

3 divides ab

But 3 does not divide a=1

b)

True

Because 3 is prime so if 3|ab then 3 must divide a or b

c)

False

Let, a=b=2

ab=4

4|ab but 4 does not divide a or b

d)

True

Because by definition of p being prime

p|ab then p|a or p|b