**Question**

**Introduction**

This is likely
an primary mathematics olympiad-type of question, but lower secondary pupils
can also try this. It involves deeper
thinking. But where do we begin? Sometimes it is good to begin from the
beginning, and then follow your nose.

**Reminders**

**Solution**

Suppose

*N*is a whole number such that 1__<__*N*__<__1000 and*N*can be expressed as*N*=

*a*

^{2}–

*b*

^{2}= (

*a*–

*b*)(

*a*+

*b*)

a difference of squares. So

*N*can be split as a product of two factors (*a*+*b*) and (*a*–*b*). Observe that (*a*+*b*) – (*a*–*b*) = 2*b*, which is an even number.
The
difference between the two factors is an even number. This can only mean that the two factors are

*both odd*or*both even*. You cannot have one of them odd and the other even, because when you subtract them, you would get an odd number. We now have three cases:-
Case
1a:

*N*is even but not divisible by 4.
Case
1b:

*N*is divisible by 4 (and, of course, is even)
Case
2:

*N*is odd i.e. both (*a*+*b*) and (*a*–*b*) are odd**Ans:**750

**Remarks**

In the
foregoing, it is possible for

We have solved the problem using logic, even-vs-odd analysis and the three important algebraic identities under reminders (highlighted in orange). We also used the special cases (highlighted in light blue) and made observations based on them.

*b*to be zero. 0 happens to be a perfect square, because 0^{2}= 0. However, we need not worry about this, because the above algebra is general enough to cover the case where*b*is 0.We have solved the problem using logic, even-vs-odd analysis and the three important algebraic identities under reminders (highlighted in orange). We also used the special cases (highlighted in light blue) and made observations based on them.

H04. Look for
pattern(s)

H05. Work
backwards

H09. Restate
the problem in another way

H11. Solve part
of the problem

H12* Think of a
related problem

H13* Use
Equation / write a Mathematical Sentence

**Suitable Levels**

*****Primary School Mathematics Olympiad

*****Secondary 2 Mathematics » grade 8 (expansion and factorisation)

* anyone who is game for a challenge in algebra and number theory

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