Introduction to Modular Arithmetic, Part 3b
See all posts in this series In this post we'll look at some applications of the result from last time that we gave the non-standard name Modular Arithmetic Consistency Theorem , or MACT for short. In particular, we will finally have a proper look at powers of integers. Theorem (MACT) Let k be a positive integer, and let a_1, b_1, a_2 and b_2 be integers. Suppose that a_1\equiv a_2~~(\mathrm{mod}~k) and b_1\equiv b_2~~(\mathrm{mod}~k). Then a_1+b_1 \equiv a_2+b_2 ~~(\mathrm{mod}~k)~~~ \textrm{and}~~~ a_1b_1 \equiv a_2b_2 ~~(\mathrm{mod}~k)\,. Warning! You can't expect anyone else to know what this so-called MACT is, so i f you want to use the name MACT in your work, you will need to explain which result it is that you are calling the MACT. (However you can usually use it implicitly in your working without actually naming it. The standard official name might be a bit long: Consistency of the equivalence relation congrue...