2 5pt Recall that in the lecture we will provehave proved th

2. (5pt) Recall that in the lecture we will prove/have proved the following property of prime numbers: If p is a prime number and a, b are arbitrary integers then (1) p a. b Recall also that in the discussion you proved, using strong induction on n, the following generalization of (1): If pis prime number and a1, a2 ...,an are arbitrary integers then (2) play a2 ...an p ak for some k such that 1 S k S n Notice the resemblance between (2) and (1): Statement (2) can be written as (plan v plaz v v plan) (3) Prove (2) using the least counterexample argument: Assume (2) is false for some positive integer n, some prime p and some integers a1, a2 ...,an, then look at the smallest such n and derive a contradiction

Solution

So we have p | a.b => p | a or p | b which is true because p is a prime number which doesn\'t have factors other than 1 and p itself. So if p divides (a.b) either or them should be a multiple of p (both of them being multiples also considers that either of them being a multiple).