Prove the square root of 7 is irrational using the Division Algorithm and case reasoning I proved this previously using proof by contradiction like so:

I am not seeing where to start to prove it using the Quotient Remainder theorem or case reasoning however. Can anyone see the best way to go about doing this? Thanks!


If the use of division algorithm is not compulsory, then a simple argument using prime factorization theorem will suffice to prove that $7|a^2 \implies 7|a$. 
Consider the contra-positive statement $7\nmid a \implies 7\nmid a^2$. The fact that $7$ does not divide $a$ means that $a$ does not have $7$ as its prime factor. Thus $a^2$ will not have any $7$ as prime factor too. Hence $a^2$ is not divisible by $7$ either.
