Question
Asked Feb 12, 2019
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Prove or disprove: Let a and b be integers and let p be a prime number. if p | ab then p | a or p | b.

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Expert Answer

Step 1

The above statement is termed as Euclid’s Lemma.

We will prove it in two ways: -

First one: -

Let us consider that ‘p’ is a prime that divides ‘ab’, but ‘p’ not divide one of the integers say ‘a’.

Then we need to show that it must divide ‘b’.

As it has been given that ‘p’ is a prime and we have considered that ‘p’ does not divide ‘a’, then ‘a’ and ‘p’ are prime to each other with some relation and ‘gcd (a, p) =1’ and from the theorem of ‘gcd’ as a linear combination.

For any integers ‘a’ and ‘b’ of non-zero value there must be two integers ‘s’ and ‘t’ for which gcd(a,b) = ‘as+bt’.

Also, the smallest positive integer of as+bt form is gcd(a,b).

 

 

Thus, two integers s and t will be there which has relations as 1=as+pt.

So, we have abs+ptb = b, and as p divides the left-hand side of the obtained equation, p also divides b. 

Step 2

Second way to proof this statement.
Consider the contradiction at the start:

Let us consider that ‘p’ will not divide ‘a’ and also ‘p’ does not divide ‘b’, but ‘p’ divides ‘ab’.

So, we have gcd(p,a)=1and ...

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