Please Login to access more options.
Problem 25 (Modular Arithmetic Properties)
Let $n$ be a positive integer. Let $a,b\in \mathbb{Z}$.
- Show that $((a \mod n)+(b\mod n))\mod n = (a+b)\mod n$. In other words, show that the sum of the remainders has the same remainder as the original sum.
- Show that $((a \mod n)(b\mod n))\mod n = (ab)\mod n$. In other words, show that the product of the remainders has the same remainder as the original product.
Remember, you'll want to use the division algorithm to obtain quotient and remainders when dividing $a$, $b$, $ab$, and $a+b$ by $n$. You'll need to use the problem, Remainders Equal Iff Difference Is A Multiple, to complete your work.
The following pages link to this page.