I am a perfect square that is 11 more than a perfect square and 13 less than a perfect square. Which number am I?
To find the number that satisfies the given conditions, let's break down the problem step by step.
Step 1: Start by identifying the conditions given in the problem:
- You are a perfect square.
- You are 11 more than another perfect square.
- You are 13 less than another perfect square.
Step 2: Express the conditions as equations:
- Let's use the variable "x" to represent the perfect square number that you are 11 more than.
- Let's use the variable "y" to represent the perfect square number that you are 13 less than.
Based on this, we can write the following equations:
1) You = x^2 (You are a perfect square)
2) You = x^2 + 11 (You are 11 more than another perfect square)
3) You = y^2 - 13 (You are 13 less than another perfect square)
Step 3: Set the equations equal to each other:
Since "You" (the number we are looking for) is equal to both x^2 + 11 and y^2 - 13, we can set these two expressions equal to each other:
x^2 + 11 = y^2 - 13
Step 4: Rearrange the equation:
To solve for "y" in terms of "x", we need to bring all terms involving "y" to one side of the equation:
y^2 = x^2 + 11 + 13
y^2 = x^2 + 24
Step 5: Find a perfect square number that is 24 greater than x^2:
Now, let's find a perfect square number that is 24 greater than x^2. We can start from x = 1 and calculate the squares until we find one that works.
x = 1: x^2 = 1^2 = 1, but 1 + 24 = 25, which is not a perfect square.
x = 2: x^2 = 2^2 = 4, but 4 + 24 = 28, which is not a perfect square.
x = 3: x^2 = 3^2 = 9, but 9 + 24 = 33, which is not a perfect square.
x = 4: x^2 = 4^2 = 16, and 16 + 24 = 40, which is not a perfect square.
After checking a few more numbers, we find:
x = 5: x^2 = 5^2 = 25, and 25 + 24 = 49, which is a perfect square (7^2).
So, the number you are is 49 (7^2).
To recap:
The number that satisfies the given conditions is 49, which is 11 more than 36 (6^2) and 13 less than 64 (8^2).