n and m are positive integers that satisfy n^3+2n^2=m^2. If 1≤n≤1000, how many possible pairs of (n,m) are there?
To find the number of possible pairs of (n, m) that satisfy the given equation, we need to check all values of n from 1 to 1000 and see if there is an integer value of m that satisfies the equation for each value of n.
Let's go through the process step by step:
1. Start by initializing a counter variable to keep track of the number of possible pairs. Set it to 0.
2. Now, iterate through all the values of n from 1 to 1000.
3. For each value of n, calculate n^3 + 2n^2.
4. Check if the result of n^3 + 2n^2 is a perfect square (i.e., an integer value for m^2). You can do this by finding the square root of n^3 + 2n^2 and checking if it is an integer.
5. If the square root of n^3 + 2n^2 is an integer, increment the counter variable by 1.
6. Continue the iteration until you have checked all the values of n from 1 to 1000.
7. Once the iteration is complete, the counter variable will hold the number of possible pairs of (n, m) that satisfy the given equation.
Implementing this algorithm in code will give you the desired result.