Q:

Prove: If n is a positiveinteger and n2 isdivisible by 3, then n is divisible by3.

Accepted Solution

A:
Answer:If [tex]n^2[/tex] is divisible by 3, the n is also divisible by 3.Step-by-step explanation:We will prove this with the help of contrapositive that is we prove that if n is not divisible by 3, then, [tex]n^2[/tex] is not divisible by 3.Let n not be divisible by 3. Then [tex]\frac{n}{3}[/tex] can be written in the form of fraction [tex]\frac{x}{y}[/tex], where x and y are co-prime to each other or in other words the fraction is in lowest form.Now, squaring[tex]\frac{n^2}{9} = \frac{x^2}{y^2}[/tex]Thus, [tex]n^2 = \frac{9x^2}{y^2}[/tex][tex]\frac{n^2}{3} = \frac{3x^2}{y^2}[/tex]It can be clearly seen that the fraction [tex]\frac{3x^2}{y^2}[/tex] is in lowest form.Hence, [tex]n^2[/tex] is not divisible by 3.Thus, by contrapositivity if [tex]n^2[/tex] is divisible by 3, the n is also divisible by 3.