PROOFSIN MATHEMATICS 303
File Name : C:\Computer Station\Maths-IX\Chapter\Appendix\Appendix– 1 (03– 01– 2006).PM65Since x and y are even, they are divisible by 2 and can be expressed in the form
x = 2m, for some natural number m and y = 2n, for some natural number n.
Then xy = 4 mn. Since 4 mn is divisible by 2, so is xy.
Therefore, xy is even. Theorem A1.3 :The product of any three consecutive even natural numbers is
divisible by 16.
Proof :Any three consecutive even numbers will be of the form 2n, 2n + 2 and
2 n + 4, for some natural number n. We need to prove that their product
2 n(2n + 2)(2n + 4) is divisible by 16.
Now, 2n(2n + 2)(2n + 4) = 2n × 2(n +1) × 2(n + 2)
= 2 × 2 × 2n(n + 1)(n + 2) = 8n(n + 1)(n + 2).
Nowwe have two cases. Either n is even or odd. Let us examine each case.
Suppose n is even : Then we can write n = 2m, for some natural number m.
And, then 2n(2n + 2)(2n + 4) = 8n(n + 1)(n + 2) = 16m(2m + 1)(2m + 2).
Therefore, 2n(2n + 2)(2n + 4) is divisible by 16.
Next, suppose n is odd. Then n + 1 is even and we can write n + 1 = 2r, for some
natural number r.
We then have : 2 n(2n + 2)(2n + 4) = 8 n(n + 1)(n + 2)
= 8(2r – 1) × 2r × (2r + 1)
= 16r(2r – 1)(2r + 1)Therefore, 2n(2n + 2)(2n + 4) is divisible by 16.
So, in both cases we have shown that the product of any three consecutive even
numbers is divisible by 16.
We conclude this chapter with a few remarks on the difference between how
mathematicians discover results and how formal rigorous proofs are written down. As
mentioned above, each proof has a key intuitive idea (sometimes more than one).
Intuition is central to a mathematician’ s way of thinking and discovering results. Very
often the proof of a theorem comes to a mathematician all jumbled up. A mathematician
will often experiment with several routes of thought, and logic, and examples, before
she/he can hit upon the correct solution or proof. It is only after the creative phase
subsides that all the arguments are gathered together to form a proper proof.
It is worth mentioning here that the great Indian mathematician Srinivasa
Ramanujan used very high levels of intuition to arrive at many of his statements, which