# What is a least residue?

## What is a least residue?

The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one representative of each residue class modulo n. For example. the least residue system modulo 4 is {0, 1, 2, 3}.

## What is the divisibility theorem?

A test for divisibility is called Casting Out Nines: Theorem. A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. Essentially the same reasoning shows: Page 8 Theorem. A positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

Which is the least positive residue of N?

The least positive residue of $a$ modulo $n$ is the smallest positive integer $k$ such that $a\\equiv k\\pmod n$. Similarly (and more commonly), the least nonnegative residue of $a$ modulo $n$ is the smallest nonnegative integer $k$ such that $a\\equiv k\\pmod n$; they are the same except when $a$ is a multiple of $n$.

### Which is the least positive residue in modular arithmetic?

I just learned modular arithmetic and my book doesn’t explain what least positive residues are so I’m a bit lost. The least positive residue of a modulo n is the smallest positive integer k such that a ≡ k ( mod n).

### What is the divisibility of a prime number?

Number Theory Divisibility and Primes Deﬁnition. If a and b are integers and there is some integer c such that a = b·c, then we say that b divides a or is a factor or divisor of a and write b|a. Deﬁnition (Prime Number).A prime number is an integer greater than 1 whose only positive divisors are itself and 1.

Which is an example of a positive divisor?

Example 1 The number 102 has the positive divisors 1, 2, 3, 6, 17, 34, 51, 102, and the number 170 has the positive divisors 1, 2, 5, 10, 17, 34, 85, and 170. The common positive divisors are 1, 2, 17, and 34. Hence (102; 170) = 34.