Square Numbers

What is a Square Number?

A natural number $n$ is called a perfect square (or square number) if it is the square of some natural number $m$. That is, $n = m^2$.

Geometrically, a perfect square represents the area of a square with integer side lengths.

Properties of Perfect Squares

1. The Unit Digit Rule

Look at the squares of numbers from 1 to 10: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$.

Notice the digits in the units place:

• Possible ending digits: 0, 1, 4, 5, 6, 9
• Impossible ending digits: 2, 3, 7, 8

2. Number of Zeros

Consider numbers ending in zero:

• $10^2 = 100$ (1 zero $\to$ 2 zeros)
• $200^2 = 40000$ (2 zeros $\to$ 4 zeros)

A perfect square must have an even number of zeros at the end. A number ending in 3 zeros (like 1000) cannot be a perfect square.

3. Odd vs Even

• The square of an even number is even ($4^2 = 16$).
• The square of an odd number is odd ($5^2 = 25$).