Consider the equation 𝑐 equals the square root of 𝑎 plus 𝑏. 𝑎 and 𝑏 are square numbers less than 100, and 𝑐 is a positive integer. Calculate two possible sets of values for 𝑎, 𝑏, and 𝑐.
Let’s start by writing down what we know. 𝑐 equals the square root of 𝑎 plus 𝑏. Another way to write this would be 𝑐 squared equals 𝑎 plus 𝑏. 𝑎 and 𝑏 are square numbers, which means they’re the result of multiplying an
integer by itself. The square numbers we’re looking for for 𝑎 and 𝑏 are less than 100. If we take the integer one and we square it, it equals one. One is a square number. Two squared equals four, so four is the next square number. Three squared gives us the square number of nine. Four squared gives us a square of 16. Five squared equals 25. Six squared equals 36. Seven squared equals 49. Eight squared equals 64. And nine squared equals 81.
If we go up any further and try 10 squared, we get 100. And we’re only looking for values less than 100 for 𝑎 and 𝑏, so we can remove 10
squared from this list. We’re also told something else. We know that 𝑐 is a positive integer. What does that mean for 𝑐 squared? It’s also a square number. And that means what we’re really looking for here is two square numbers that when
added together yield another square number. If we added one and four together we would get five, but five is not a square
number. Nine plus four equals 13, but 13 is not a square number. Nine plus 16 equals 25, and 25 is a square number.
Remember, 𝑐 squared has to equal 𝑎 plus 𝑏. We can plug in nine for 𝑎 and 16 for 𝑏. Nine plus 16 equals 25, and 25 equals five squared. Our 𝑐 value would be five, 𝑎 would be nine, and 𝑏 would be 16. Now we go back to our square numbers and see if we can choose a second set. If we look at the values 36 and 64 together, when we add them they equal 100. 100 is a square number. 100 is 10 squared, and that makes the 𝑐 value in this case the positive integer
10. If 𝑐 equals 10, 10 squared equals 36 plus 64. For this set, 𝑎 equals 36, 𝑏 equals 64, and 𝑐 equals 10.