Download Euclidean division and the greatest common divisor

The division algorithm and the greatest common divisor
Let a be an integer number and n be a natural number. Then we can find
another integer number u as well as a nonnegative integer number f < n so
that:
a = nu + r
This property is known as Euclidean division, u is known as the result of
the division and r as the remainder.
Let’s take a look at some examples:
• When dividing 20 by 3, we get the equation:
20 = 3 × 6 + 2
so the result is 6 and the remainder is 2.
• When dividing −20 by 3, we get the equation:
−20 = 3 × (−7) + 1
so the result is −7 and the remainder is 1.
If two integer numbers a and b have the same remainder when divided by
the natural number n, then they are called congruent modulo n. and we
write:
a ≡ b mod n
This is the same as requiring that the difference a − b is divisible by n.
We now turn to the study of the greatest common divisor of two integer
numbers. Let a and b be two integer numbers with at least one them nonzero. The greatest common divisor of a and b denoted by gcd(a, b) is the
largest natural number that divides both a and b.
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The most important property of the greatest common divisor of two integer
numbers is that it can be written as an integral combination of the two
numbers. More precisely, one can find integer numbers x and y such that:
gcd(a, b) = ax + by
There is a very interesting algorithm based on Euclidean division that allows
us to determine both the greatest common divisor and the numbers x and
y that appear on the previous equation.
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