The biggest use of euclidean division is to find the greatest common divisor of two numbers.

Use (a, b) to represent the greatest common divisor of a and b. There is a theorem: It is known that a, b, c are positive integers. If a is divided by b and the remainder is c, then (a, b) = (b, c). (Please refer to other materials for the proof process)

Example: Find the greatest common divisor of 15750 and 27216.

Solution:

∵27216=15750×1+11466 ∴(15750, 27216)=(15750, 11466)

∵15750=11466×1+ 4284 ∴(15750, 11466)=(11466,4284)

∵11466=4284×2+2898 ∴(11466,4284)=(4284, 2898)

∵4284= 2898×1+1386 ∴(4284,2898)=(2898,1386)

∵2898=1386×2+126 ∴(2898,1386)=(1386,126)

∵1386=126×11 ∴ (1386, 126) = 126

So (15750, 27216) = 216

The euclidean division method is more suitable for finding two larger ones The greatest common divisor of numbers.

Extended information;

The euclidean algorithm, also known as the Euclidean algorithm, is a method of finding the greatest common divisor. Its specific method is: divide a larger number by a smaller number, then divide the divisor by the remainder (the first remainder), then divide the first remainder by the remainder (the second remainder), and so on until Until the final remainder is 0. If you are looking for the greatest common divisor of two numbers, then the final divisor is the greatest common divisor of the two numbers.

Another way to find the greatest common divisor of two numbers is the subtraction method.

The greatest common divisor of two numbers refers to the largest positive integer that can divide them simultaneously.

Suppose the two numbers are a and b (a≥b), the steps to find the greatest common divisor of a and b are as follows:

(1) Divide a by b (a≥ b), got?.

(2) If ?, then ?;

(3) If ?, then divide b by ? to get ?.

(4) If ?, then ?; if ?, continue to divide ? by ?,...and so on until it can be divided.

The divisor whose last remainder is 0 is the greatest common divisor of ?