Calculators are for fools!

$\displaystyle 87654567 \times 87654569 + 1$ $\displaystyle =87654567 \times (87654567 + 2) + 1$

Let $\displaystyle n=87654567$ , then the above expression becomes:

$\displaystyle =n(n+2)+1=n^{2} + 2n + 1 = (n+1)^{2}$ , which is clearly a perfect square.

Except $\displaystyle 1$ , every perfect square is a composite number. Therefore $\displaystyle 87654567 \times 87654569 + 1$ is a perfect square.

We also infer that if a natural number can be expressed as the product of two natural numbers differing by 2, then 1 more than such a number will always be a perfect square and that too, of the number between the two numbers differing by 2.

Another solution:

$87654567 \times 87654569 + 1 = (87654568-1)(87654568+1) + 1 = 87654568^{2} - 1^{2} + 1 = 87654568^{2}$ , which is a perfect square.

Let x = 87654568, the equation would become: (x-1)(x+1)+1 = ( $x^{2}$ - 1) + 1 = $x^{2}$

I can see and understand the mathematical reasons for why the answer is 'perfect square' BUT I did not use that in my solution.

I looked at and odd x odd +1

we know that:

odd x odd = odd

therefore:

odd x odd +1 = even

and except for 2, no even number is a prime number, so by the process of elimination the answer is perfect square