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I. Euclidean division and divisibility criteria
1. euclidean division
To perform the Euclidean division of a whole number (the dividend) by a non-zero whole number (the divisor) is to find two whole numbers, the quotient and the remainder, such that: 
with remainder < divisor
Example:
2. Multiples and divisors
The remainder of the Euclidean division of 36 by 9 is 0. Indeed: 36 = 4×9 + 0.
We say that “36 is a multiple of 9”, “36 is divisible by 9”, “9 is a divisor of 36”.
3. Criteria of divisibility by 2, 3, 4, 5, 9 and 10
An integer is divisible by :
- 2 when it ends in 0, 2, 4, 6 or 8;
- 3 when the sum of its digits is a multiple of 3;
- 4 when the number formed by its last two digits is divisible by 4;
- 5 when it ends with 0 or 5;
- 9 when the sum of its digits is a multiple of 9;
- 10 when it ends with 0.
Examples:
- 240 is divisible by 2 because it ends in 0 but also by 5 and 10;
- 65 is divisible by 5 because it ends in 5;
- 1845 is divisible by 9 because 1+8+4+5=18 and 18 is a multiple of 9;
- 128 is divisible by 4 because 28 is divisible by 4 in effect .
II. Decimal division
Let a be a decimal number and b an integer other than zero.
To perform the decimal division of a by b is to find the number called quotient
by which to multiply b to obtain a :
and .
Example 1:
Perform the decimal division of 9.2 by 4.
The quotient of 9.2 by 4 is the decimal number 2.3.
Thus that is to say that .
Example 2:
Perform the decimal division of 8 by 3.
This division never ends.
The quotient of 8 by 3 is not a decimal number.
In this case, we can give an approximate value of the quotient :
- (to the nearest tenth).
- (to the nearest hundredth).
Have you assimilated the course on euclidean and decimal division in 6th grade?
Un QCM de maths sur la division euclidienne et décimale en 6ème.
Skills to learn about Euclidean and decimal division:
- Know the definition of a dividend, divisor, quotient and remainder;
- Know how to do Euclidean division with integers;
- Know how to do decimal division;
- Give an order of magnitude of a quotient;
- Know how to divide by 10, 100, 1000, 0.1, 0.01, 0.001 etc…
This course is in accordance with the official programs of thenational education.
In addition, you can continue with the exercises on the 4 operations in the sixth grade.
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