Arithmetic is a very important third grade math course. In addition, you will see the definition and property of Euclidean division as well as the definition of a prime number and the prime factor decomposition theorem of any integer.
The student will need to know the definition of a divisor and a multiple and know the different criteria for divisibility. Next, you will develop arithmetic skills with the prime factor decomposition of an integer.

We will end this chapter on arithmetic by solving real-life problems in third grade.

## I. Euclidean division in arithmetic :

1.euclidean division :

Definition:

Consider two positive integers a and b with b nonzero and a>b. Performing the Euclidean division of a by b is to find theunique pair of positive integers (q,r) such that :

with .

If r=0, we say that a is a multiple of b or that b is a divisor of a.

Example:

Let us take a=187 and b=13, we apply the Euclidean division to obtain q and r.

So with 5<13.

### 2. multiples and divisors in arithmetic :

Example:

Let’s take a= 135 and b = 15.

We have .

So 135 is a multiple of 15 and 15 is a divisor of 135.

Remarks:

• An integer has a finite number of divisors, but an infinite number of multiples.
• An integer greater than 1 always has at least two divisors: 1 and itself.

### 3.criteria of divisibility with arithmetic :

Ownership:

Consider a positive non-zero integer n.

• n is divisible by 2 if it ends in 0,2,4,6, or 8.
• n is divisible by 5 if it ends in 0 or 5.
• n is divisible by 3 if the sum of its digits is a multiple of 3.
• n is divisible by 9 if the sum of its digits is a multiple of 9.

Example:

• 915 is not divisible by 2 because it ends in 5.
• 915 is not divisible by 4 because 15 is not.
• 915 is divisible by 3 because and 15 is divisible by 3.

## II. Prime numbers with arithmetic:

1.Definition:

Definition:

Consider a positive non-zero integer n. The integer n is a prime if, and only if, it has exactly two divisors which are 1 and itself.

Examples:

• The list of prime numbers less than 100 : 2,3,5,7,11,13,17,19,23,29,31,37.
• 91 is not a prime number because so it has 4 divisors.

### 2. decomposition into prime factors :

Ownership:

Consider a positive integer n greater than 1.The integer n can be written as a product of primes.

We have , this writing, called the prime factor decomposition of n, is unique, to the order of the factors.

Examples:

Ownership:
To decompose an integer into a product of prime factors, we must progressively decompose the integer using prime numbers in ascending order.

Example:

We want to decompose the integer 3,626 into a product of prime factors.

### 3. Irreducible fractions :

Definition:

A fraction is irreducible when it cannot be simplified.

The fraction is irreducible if, and only if, the greatest common divisor, noted gcd(a,b), of the numbers a and b is 1.

Remark:

A fraction is irreducible when the greatest common divisor of a and b (noted gcd(a,b)) is 1.

Example:

where is an irreducible fraction because gcd(12,259)=1.

## Have you assimilated the course on arithmetic and decomposition into prime factors in 3rd grade?

L'arithmétique et la décomposition en facteurs premiers

QCM de maths sur l’arithmétique et la décomposition en facteurs premiers en 3ème.

Cette publication est également disponible en : Français (French) العربية (Arabic)

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