1998 CEMC Gauss (Grade 7) Problems/Problem 18

Problem

The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown: \[1. \quad AUSSG \qquad 9981\] \[2. \quad USSGA \qquad 9819\] \[3. \quad SSGAU \qquad 8199\] If the pattern continues in this way, what number will appear in front of GAUSS 1998?

$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$

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 $\text{(E)}.$