7 Recall that if p divides a we write this as pIa Let p be a

7. Recall that if p divides a we write this as pIa. Let p be a prime number and a and b

positive integers, such that pIab. Prove that we must have either pIa or pIb

Solution

We prove by Contrapositive method

we assume that p doesnt divide \'a\' -------(1)

ie p is not a factor of \'a\' --------(2)

simi;\\larly p doesnt divide \'b\' ----(3)

ie p is not a factor of \' b\' ----(4)

from 1 and 3 p is not a factor of \' ab\'

from 2 and 4 p doesnt divide \' ab \' which contradicts to the given fact that p divides ab

hence our assumption is wrong ie either 1 or 3 is false

therfore either p divides \'a\' or \'b\'