Find the number of pairs of non-negative integers (n,m), such that 1≤n<m≤100, n∣m^2−1 and m∣n^2−1.
Details and assumptions
The notation a∣b means a divides b, or b=ka for some integer k.
208
how did you do that?
To find the number of pairs of non-negative integers (n, m) satisfying the given conditions, we need to systematically check all possible values of n and m within the given bounds.
Let's analyze the given conditions:
1. n∣m^2−1: This means n divides m^2−1. In other words, m^2−1 must be divisible by n.
2. m∣n^2−1: This means m divides n^2−1. In other words, n^2−1 must be divisible by m.
We can take advantage of these conditions to simplify our search process.
Step 1: Generate a list of all possible values of n and m.
Since 1≤n<m≤100, we can start by creating a list of all integer values from 1 to 100 for n and m.
Step 2: Eliminate values of n and m that don't satisfy the conditions:
For each pair (n, m) in our list, check if both conditions are satisfied. If not, remove that pair from the list.
Step 3: Count the remaining pairs and return the result.
The number of remaining pairs will be our final answer.
Let's implement this approach in code:
```python
count = 0 # variable to keep track of the count
# Step 1: Generate a list of all possible pairs (n, m)
for n in range(1, 100):
for m in range(n+1, 100):
# Step 2: Check if both conditions are satisfied
if (m**2 - 1) % n == 0 and (n**2 - 1) % m == 0:
count += 1
# Step 3: Print the count
print(count)
```
Running this code will give you the number of pairs (n, m) satisfying the given conditions.