There are two parts to this question. Part a) The difference between two square numbers is 40. What could the two square numbers be? Part b) Florence says, “The difference between any two different square numbers is always even.” Is she correct? Give a reason for your answer.
In both parts of this question, we are asked to consider the square numbers. Squaring a number involves multiplying it by itself. Therefore, the square numbers are one, four, nine, 16, 25, 36, 49, 64, 81, 100, and so on. This is because one squared or one multiplied by one is equal to one. Two squared is equal to four. Three squared is equal to nine. Four squared equals 16, and so on.
We were asked to find two square numbers where the difference is 40. Adding 40 to nine gives us 49. This means that 49 minus nine is equal to 40. Two square numbers with a difference of 40 are nine and 49.
In order to answer part b), let’s consider the first five square numbers: one, four, nine, 16, and 25. Florence said that the difference between any two different square numbers is always even. In order to prove that this statement is incorrect, we just need to find one example where the difference is not even.
For example, four minus one is equal to three, which is an odd number. Other examples that disprove Florence’s statement are nine and four, as nine minus four equals five, and 25 and 16. 25 minus 16 is equal to nine. We can therefore say that, no, Florence is not correct and we must give at least one example that proves this.
We could go one stage further in this question. We know that an odd number minus an even number equals an odd number. Likewise, an even number minus an odd number gives us an odd number. As the square numbers alternate between odd and even, there will be examples where the difference between the square numbers is odd.