What is the unique three-digit positive integer satisfying 100x = 11 (mod \(997\))

Guest Jan 25, 2022

#1**0 **

Since x is a 3 digit integer, then the equation is:

\(100x = 11 + 997k\)

The smallest value of k is 11, or else x could not be a 3 digit integer satisfying the restrictions.

100x must end with two 0's, so 997k must end with 89.

Since 7 is the only number that can multiply the last digit of 997 to make the new number end with a 9.

After some testing we get k = 37.

Here is our new equation:

100x = 11 + 997 * 37

100x = 36900

**x = 369.**

proyaop Jan 25, 2022