Find the number of pairs of non-negative integers (n,m), such that 1≤n<m≤100, n∣m2−1 and m∣n2−1.
Details and assumptions
The notation a∣b means a divides b, or b=ka for some integer k.
To find the number of pairs of non-negative integers (n,m) that satisfy the given conditions, we can follow a systematic approach:
Step 1: Generate all possible values of n and m
In this case, we need to find pairs (n,m) where 1≤n<m≤100. We need to consider all possible values of n and m within this range.
Step 2: Check if n divides (m^2 - 1)
For each pair (n,m), check if n divides (m^2 - 1). If it does not, then that pair does not satisfy the condition.
Step 3: Check if m divides (n^2 - 1)
For each pair (n,m), check if m divides (n^2 - 1). If it does not, then that pair does not satisfy the condition.
Step 4: Count the valid pairs
For each pair (n,m) that satisfies both conditions, count it as a valid pair.
Now, let's go through these steps to find the number of pairs:
Step 1: Generate all possible values of n and m
We need to find pairs (n,m) where 1≤n<m≤100. To generate all possible pairs, we can use nested loops, where the outer loop iterates over the values of n from 1 to 99, and the inner loop iterates over the values of m from n+1 to 100.
Step 2: Check if n divides (m^2 - 1)
For each pair (n,m), we can check if n divides (m^2 - 1) by calculating the remainder when (m^2 - 1) is divided by n using the modulo operator (%). If the remainder is 0, then n divides (m^2 - 1).
Step 3: Check if m divides (n^2 - 1)
Similarly, for each pair (n,m), we can check if m divides (n^2 - 1) by calculating the remainder when (n^2 - 1) is divided by m. If the remainder is 0, then m divides (n^2 - 1).
Step 4: Count the valid pairs
For each pair (n,m) that satisfies both conditions, we increment a counter to keep track of the number of valid pairs.
By following these steps, we can find the number of pairs of non-negative integers (n,m) that satisfy the given conditions.