This is a job for the Chinese Remainder Theorem. You can find information about that theorem by using Google. But the numbers in your example are way too big if you have never seen such problem before. The way to learn how to solve this type of problem is to make three lists. A list of all numbers that leave 1 after division by 11:

1, 12, 23, 34, ...
(Of course, you get the next number in the list by adding 11 to the previous number.)

Then you list the numbers that leave 8 after division by 13:
8, 21, 34, 47, ...

and the numbers that leave 3 after division by 17:
3, 20, 37, 54, ...

You then have to extend your three lists until you get a match in all three. Unfortunately, with your starting numbers, each of your three lists will contain more than 100 items -- it's like playing the slot machines in Las Vegas! For better problems for beginners look at

http://mathcentral.uregina.ca/QQ/database/QQ.09.98/barker1.html

and

http://mathcentral.uregina.ca/QQ/database/QQ.09.96/tan1.html (the second part only)

and

http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/mhiko1.html

Chris