Q.1. How do I prove that the law of Syllogism is a tautology?

For Q.2 & Q.3, Prove the Theorem.

Assume that all the symbols used represent positive integers.

Q.2. If ac divides bc, then a divides b.

Q.3. For all positive integers n greater than 10,
12(n-2)<n^2-n(n square-n)

Thanks for any help

Question 1
The Law of Syllogism states that:

If you accept “If P, then Q” as true and “if Q, then R” as true, “if P, then R” must also be true. As a real life example, consider:

If I fail this assignment, my grade will drop. If my grade drops, I will not graduate with honors. Therefore, if I fail this assignment, then I will not graduate with honors.

That this is a tautology may be proved from a truth table. An example of such a table, with a definition of tautology and the law of syllogism, may be found here.

Question 2
ac divides bc
therefore ac*q = bc where q = some unknown integer
therefore a*q = b (dividing both sides by c)
therefore a divides b, because q is an integer

Question 3
Perhaps some can offer a better methodology in support of the proof. However:

12n – 24 < n2 – n

adding "n" to both sides:

13n – 24 < n2

You may substitute 11 and 12 to confirm that they are true. 13 is obviously true, since 13-squared minus 24 is less than 13-squared. For any values above 13, n-squared will always be greater than 13 times n.