Regular patterns in the units digit of sets of numbers generated by some simple rule afford a great opportunity for students to exercise their research skills in pure mathematics, and get a taste of what it feels like to be a research mathematician. For those who have studied some algebra, the experience can be enhanced by asking them to explain/prove the patterns they have found, using simple algebraic reasoning. In this article, we will explore units digits in multiples of each of the numbers from 1 to 10. Later articles will explore sums of the digits, as well as units digits for powers of a given number under 10, and units digits in the sequence of squares, in the sequence of cubes, and higher powers. These lead to more challenging and unexpected results, most of which can be explained by simple reasoning. Teachers should guide students to find and explain these results themselves, giving help when necessary. Especially useful for gifted students.