What is the largest integer 𝑎 such that both one over 𝑎 and 𝑎 over the square root of 463 are between zero and one?
Here’s what we know. One over 𝑎 needs to fall between zero and one. 𝑎 over the square root of 463 must also fall between zero and one. We want the largest integer that keeps both of these statements true. First, let’s think about 𝑎 over the square root of 463.
If 𝑎 is equal to the square root of 463, that wouldn’t work because then the fraction would be equal to one. That means that our number 𝑎 is the integer that is less than the square root of 463. For example, 20 squared equals 400 and that’s less than 463. Is there an integer larger than 20 whose square value is still less than 463?
We can try 21 and square that integer. 21 squared is equal to 441. 441 is still less than 463. Let’s go up one more, 22 squared. 22 squared is 484. It’s too large. 21 is the largest integer value that keeps this statement true.
Now, we need to take our integer 21 and substitute it into the equation on the left. Is one over 21 between zero and one? It is. In the equation on the left, the larger the integer in the denominator, the closer to zero this value becomes. However, we’re limited on how large value 𝑎 can be by the equation on the right. And that means 21 is the largest integer that satisfies both of these equations.