Modular Arithmetic

Introduction to “Mods” Modular arithmetic is just a fancy phrase for a subject dealing with remainders. Before we get into this, please note that we’re restricting the dividend, divisor, and remainder to integral (which means they are also rational and real) values.

Introduction to “Mods” Contd. When we say 𝑎 𝑚𝑜𝑑 𝑚= 0, we mean that 𝑚|𝑎, or “m evenly divides a”. When we say 𝑎 𝑚𝑜𝑑 𝑚 = 1, we mean that 𝑚 | (𝑎−1) Alternatively, it means that when a is divided by m, it shall leave a remainder of 1.

Introduction to the “Arithmetic” But note that if 𝑚 | (𝑎−1), then 𝑚|(𝑎+𝑚−1) To take it a step further, we can say 𝑚|(𝑎+𝑥𝑚−1) for any integer x (whether positive or negative). This is a simple application of modular arithmetic With this information, we can confidently say that when 𝑎 = 𝑏 𝑚𝑜𝑑 𝑚, we know that 𝑎 = 𝑏 + 𝑥𝑚 for some integer x.