A conjecture is basically a statement, e.g. n2 + n + 1 is a prime number. To prove this wrong, all you need is a 'counter example'. So you could try putting things into the formula until you get something that proves it wrong.
E.g. n = 1 ⇒ n2 + n + 1 = 3 and 3 is prime.
n = 2 ⇒ n2 + n + 1 = 7 and 7 is prime.
n = 3 ⇒ n2 + n + 1 = 13 and 13 is prime.
n = 4 ⇒ n2 + n + 1 = 21 and 21 is not prime.
So now we've created a counter example and the question is complete.
Proof By Exhaustion
Some of the conjectures can be proved by testing every case. E.g.
97 is a prime number.
You can easily show that none of the factors between 2 and 96 are factors of 97 and so is indeed a prime number - but think for a minute, you don't have to try all of them - only the ones less than √97 (from 2 to 9).
97 ÷ 2 = 48.5 No.
97 ÷ 3 = 32.5 No.
97 ÷ 4 = 24.25 No.
97 ÷ 5 = 19.4 No.
97 ÷ 6 = 16.6666... No.
97 ÷ 7 = 13.857... No.
97 ÷ 8 = 12.125 No.
97 ÷ 9 = 10.777... No.
So 97 must be prime.
Proof By Deduction
In some situations like above, you might have to test hundreds, thousands or even millions of times because you can prove it wrong or right, so this is where proof by deduction comes into play. E.g.
There is a trick for squaring numbers which, like 5.5, are [a whole number +0.5].