Attached is an image of a calculus problem I submitted for a grade. The teacher sent it back with the response at the bottom in red. Would someone be able to explain how and what I did wrong?

Select a positive two-digit integer N other than 27 and 64, and do the following: A. Prove 3\\ N is not a rational number. Justify your work. 31 10 Proof by contradiction So we will start by assuming that 31 10 is a rational number. When we assume that r uus a rational number, that means that there exists integers M and N with with N greater than 0 such that we can set So we will let V 10 = where m and n are integers in lowest terms (with n not being equal to 0 since it is in the denominator) Then we will take 3rd exponent of both sides and get: 10 = m Then we will multiply both sides by n > and get: 10n = m 10 is a factor of m Now we can ask if m is divisible by 13. In general we can say no. Below is a counter example: 10n 3 and m=4 10n = 4 3 10 does not divide 4. So now we are back to 10n 3 = m 3 Euclid's Lemma: states that if a prime p divides a product of integers ab, then p divides a or b. Since 10 is not prime, we can use Application/Corollary of Euclid's Lemma which states is prime number p divides the x , then p divides x. 2 * 5 * n 3_ 3 m 5 is a prime numbers and divides m " so by using the application of Euclid's Lemma, 5 divides m. Since 5 I m then m=5k, KEZ 2 * 5 * n = (5k) 2 *5 * n 3 = 5 3 2 * n 3 - 5 3x 3 Since 5 does not divide 2 then 5 In 3 Since 5 is a prime number and 5 divides by n , so by using application of Euclid's lemma 5 divides n. Hence 5 divides m and 5 divides n, so m, n are not coprime. Contradiction to the assumption that m, n have no common factors other than 1. Therefore, 10 is irrational number. The strategy of using a proof by contradiction to show that : 10 is not rational has merit, and appropriate assumptions are stated on m and n to begin the proof. It is accurately established that 10n3 = m3. The subsequent discussion is unclear, as it is not known how the consideration of whether m is divisible by 13 or whether 10 divides 4 advance the proof. In the discussion of n3, the statement "5 divides by n3" is unclear and not necessarily accurate