Question proof √3 is irrational and show reason.

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We prove that √3 is irrational. Assume to the contrary that √3 is rational,

that is

√3=p/q,

where p and q are integers and q≠0. Morever, letp and q have no common divisor > 1.Then

3=(p/q)² => 3q²=p² (1)

Since 3q² is odd,it follows that q² is odd. Thenp is also odd.

This means that there exists k ∈ Z such that

p=3k. (2)

Substituting (2) into (1), we get

3q = (3k)² => 3q²=9k² => q² = 3k².

Since 3k² is even it follows that q² is even. Thenq is also evenThis is a contradiction.

Note. I'm stuck with this solution and unable to understand the meaning of it. Any more simple alternative solution to reason irrational ?