Given integers R,M with M≠0, let S(R,M) denote the smallest positive integer x satisfying the congruence
Rx≡1(mod M)
if such an x exists. If such an x does not exist, put S(R,M)=0.
Each line of this text file contains a pair of space separated integers representing R and M, respectively.
Let L be the list of integers whose k-th element is the value of S(R,M), where R and M are taken from the k-th line of the text file.
Let T be the sum of all elements of L. What are the last three digits of T?
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To find the last three digits of T, we need to calculate the sum of all elements in L and then find the remainder when divided by 1000. Here's how you can do it:
1. Read the text file line by line and extract the values of R and M from each line.
2. For each pair of R and M, calculate S(R, M) as follows:
- Initialize a variable x as 1.
- Start a loop that keeps incrementing x until Rx ≡ 1 (mod M). This can be done by checking the remainder when Rx is divided by M using the modulus operator (%).
- If such an x exists, mark the current value of x as the smallest positive integer satisfying the congruence.
- If no such x exists, set S(R, M) to 0.
3. Create a list L to store the values of S(R, M) for each pair of R and M.
4. Calculate the sum of all elements in L, denoted as T.
5. Take the remainder of T divided by 1000 to get the last three digits of T.
Let's go through an example:
Suppose the text file contains the following lines:
```
2 5
3 7
4 9
```
- For the first line, R = 2, M = 5. Calculate S(2, 5) by finding the smallest positive integer x such that 2x ≡ 1 (mod 5). In this case, 2 * 3 ≡ 1 (mod 5), so S(2, 5) = 3.
- For the second line, R = 3, M = 7. No positive integer x satisfies the congruence equation 3x ≡ 1 (mod 7), so S(3, 7) = 0.
- For the third line, R = 4, M = 9. Calculate S(4, 9) by finding the smallest positive integer x such that 4x ≡ 1 (mod 9). In this case, 4 * 7 ≡ 1 (mod 9), so S(4, 9) = 7.
The list L will be [3, 0, 7], and the sum T is 3 + 0 + 7 = 10. The last three digits of T are 010.
That's how you can find the last three digits of T based on the given instructions.