Let a be a nonzero digit and b any of the digits 1, 3, 7, or 9. To a positive integer k, we alternately append a on the left and b on the right. This process yields two infinite sequences of integers, depending on which side the appending process begins. We are interested in those values of k that result in these sequences being composed entirely of composite integers. In particular, we seek the smallest such k relatively prime to the greatest common divisor of a and b.