Show that there are exactly five three-digit square numbers less than 200.
So in order to answer this question, we’re going to need to list out all of the square numbers which are three digits and less than 200. Remember, a square number is the result of multiplying an integer by itself. For example, four is a square number because it’s equal to two multiplied by two. 100 is the smallest three-digit number. And we should be familiar with 100 as 10 squared. It’s equal to 10 multiplied by 10.
Just to be certain though, we could list out the next square number below 100: nine squared, nine multiplied by nine, which is equal to 81. We’ll then continue listing our square numbers. The next one is 11 squared, 11 multiplied by 11, which is equal to 121. The next square number is 12 squared, 12 multiplied by 12, which is equal to 144. Next, 13 squared, 13 multiplied by 13, which is equal to 169.
Now you should be familiar with 11 squared and 12 squared. But you may not be familiar with square numbers larger than this. So you can use a calculator to help you work out 13 squared and then the next couple of square numbers as well. 14 squared is equal to 196. And then the next square number, 15 squared, is equal to 225. But remember, in the question, we were asked about the three-digit square numbers which are less than 200. So 225 is too big.
We can now count the three-digit square numbers that we’ve listed which are less than 200. We see that there are five three-digit square numbers less than 200. Now because this question has asked us to show that there are exactly five three-digit square numbers less than 200, we do also need to show that there are no others. So we need to include that nine squared is 81, which is too small, and that 15 squared is 225, which is too big, as part of our answer.
So we can conclude that there are indeed exactly five three-digit square numbers which are less than 200. And if we want, we can list them. These numbers are 100, 121, 144, 169, and 196.