1998 CEMC Gauss (Grade 7) Problems/Problem 18
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Problem
The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown:
![\[1. \quad AUSSG \qquad 9981\]](http://latex.artofproblemsolving.com/a/6/f/a6f58edb19c374417ed58691a39899c773cc2337.png)
![\[2. \quad USSGA \qquad 9819\]](http://latex.artofproblemsolving.com/1/7/3/173c9dc7fc07c87198127879628ed6e063393c17.png)
![\[3. \quad SSGAU \qquad 8199\]](http://latex.artofproblemsolving.com/4/b/7/4b78a41bb799de990438a05236364b2a0a8c8a1a.png)
If the pattern continues in this way, what number will appear in front of GAUSS 1998?
Solution
"GAUSS" has 5 letters, so the sequence will reset itself every 5 terms. "1998" has 4 numbers, so the sequence will reset itself every 4 terms. The LCM of 4 and 5 is 20, so the answer is 
