Expert: Vishwas K Hajirnis - 9/5/2004

Question
"My teacher didn't explain conjectures and I'm having trouble on how to solve those equations.
  Ex: The product of an odd number and an even number is ?.
  Can you help me and give me advice and in depth description?";
  You will be seeing me again.
                            Thanks!

Answer
A conjecture is essentially a guess. By observing some
examples you are supposed to make an intelligent guess.

Now let us see what can we guess when an odd number
is multiplied by an even number.
An even number is an integer which is divisible
by 2. e.g. 2,4,6,8,... are even numbers. Whereas
integers which are not divisible by 2 are called odd
integers. e.g. 1,3,5,7,... are odd integers.
In order to make a guess it is better to try some
specific values. (Of course if these values are small
then you will have to do less calculations.)
Let us consider some pairs of odd and even integers
and check the answers.
(Here '*' denotes multiplication)

   3 * 4 = 12
So we observe here
 Odd *  Even = Even

   7 * 2 = 14
Again the form is
 Odd *  Even = Even

   5 * 6 = 30
Here again
 Odd *  Even = Even

So we may conjecture that the product of an
odd integer and an even integer is even.

One should usually try a few more values before
arriving at a conjecture.

Let us consider one more example.
Suppose I start with the number 60 and try to
divide 60 by a few numbers smaller that 60.
e.g. I try to divide 60 by 1,2,3,4,5 etc.
I observe that 60 is divisible by each of these
numbers. So I might be tempted to make a
conjecture that every number less than 60
is a divisor of 60. To check the validity
of my conjecture I may try a few more numbers
less than 60 and see whether they divide 60.
e.g. I may try 6, 10, 12, 15, 20 etc. as divisors
and say that my conjecture seems to be true.
But if you want to shoot down my conjecture
all you need to do is to point a single
exception to the rule I have guessed. For
example you may point out that 7 is less than
60 but 60 is not divisible by 7. Hence
my conjecture has been demonstrated to
be wrong. The value of 7 is called
a 'counter example' in this context.

Thus we observe:
Every conjecture may not be true.
A conjecture can be refuted by giving a
counter example.

Almost all mathematical results, before they are
proved; start their life as conjectures.
Some conjectures remain unproved for many many
years.
So unless and until a conjecture is proved to be
true or a counter example is found it remains a
conjecture.

Only when a conjecture has been proved to be
true it becomes a mathematical result (theorem).

I hope this is clear.