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

 Jan 25, 2022
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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.

 

smiley

 Jan 25, 2022

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