Vishnu's MATHCOUNTS 2004 question

When discussing squares of numbers, the following question came up

When squaring numbers (see Squares.htm) we were discussing some patterns. Comparing pairs like 50 and 60 or 60 and 70 it seems that the difference between their squares is the sum of the first digits multiplied by 100. Vishnu asked if this is true?

At first, I could see that it appears to work.

50 * 50 = 2500 and 60*60=3600        The difference, 3600 - 2500  is 1100 which is (5 + 6) * 100.  Of course, you are probably noticing that 60 = 6 * 10 and 70 = 7 * 10.

Since 70 * 70 = 4900 and 4900 - 3600 = 1300 it also works that 1300 is (6+7) * 100.

But when Vishnu asked I didn't see "why" while I was at the board.

Later, I looked more closely at the problem and here is the solution:

When dealing with these kinds of pairs, note that the numbers (consecutive multiples of 10) can be written in the form and which is . These are the numbers we then square.

The difference in the two squares then is .

This can also be written .

Remember that the original question asked if we could add the first digits and then multiply by 100.

and that leads to

So yes, Vishnu you are correct.