For integers `a` and `b`, denote by `Div`(`a`, `b`) the collection of common divisors, which are the natural numbers that divide both `a` and `b`. For example, 3 ∈ `Div`(102, 45) because 102 = 34×3 and 45 = 15× 3.

Starting from `a` = 102 and `b` = 45, we carry out the following series of divisions (called euclidean algorithm)

102 = 2×45 + 12,

45 = 3×12 + 9,

12 = 1×9 + **3**,

9 = 3×**3**.

The first equality implies that

102 and 45 are divisible by `c` ⇔ 45 and 12 are divisible `c`.

Thus `Div`(102, 45) = `Div`(45, 12). In general, it is easy to see that if `a` = `qb` + `r`, then `Div`(`a`, `b`) = `Div`(`b`, `r`). Then the series of divisions tells us

`Div`(102, 45) = `Div`(45, 12) = `Div`(12, 9) = `Div`(9, **3**) = {1, 3},

where the last equality is due to the fact that 9 is a multiple of 3, so that `Div`(9, **3**) consists of divisors of 3.

The biggest number in `Div`(`a`, `b`) is denoted `gcd`(`a`, `b`) and is called the greatest common divisor between `a` and `b`. The euclidean algorithm above shows that `gcd`(102, 45) = 3. Since the algorithm produces smaller and smaller natural numbers, it always stops (say at (`a'`, `b'`)), when one number is an integer multiple of the other (say `a'` = `qb'`). Then `gcd`(`a`, `b`) = `gcd`(`a'`, `b'`) = `b'`. In particular, we conclude that for integers, the greatest common divisor always exists.

We may introduce similar concepts for polynomials.

The greatest common divisor between two polynomials `p`(`t`) and `q`(`t`) is a polynomial `d`(`t`) = `gcd`(`p`(`t`), `q`(`t`)) with the following properties:

`d`(`t`) divides`p`(`t`) and`q`(`t`).`c`(`t`) divides`p`(`t`) and`q`(`t`) ⇒`c`(`t`) divides`d`(`t`).

Since polynomials can also be divided, the euclidean algorithm can also be carried out for polynomials. By the same reason, the algorithm produces the greatest common divisor between two polynomials. In particular, this proves the existence of the `gcd` for polynomials.

Example From this example, we already know

`t`^{4} + `t`^{3} + 3`t`^{2} + `t` + 1 = (`t`^{2} + 2`t` + 4)(`t`^{2} -` t` + 1) + (3`t` - 3).

Further computations give us

`t`^{2} -` t` + 1 = (`t`/3)(3`t` - 3) + 1,

3`t` - 3 = (3`t` - 3) 1.

Thus `gcd`(`t`^{4} + `t`^{3} + 3`t`^{2} + `t` + 1, `t`^{2} -` t` + 1) = 1.

If both `d`_{1}(`t`) and `d`_{2}(`t`) satisfy the conditions for `gcd`, then `d`_{1}(`t`) = `u`(`t`)`d`_{2}(`t`) and `d`_{2}(`t`) = `v`(`t`)`d`_{1}(`t`) for some polynomials `u`(`t`) and `v`(`t`). This implies `u`(`t`)`v`(`t`) = 1, so that both `u`(`t`) and `v`(`t`) are actually nonzero numbers. Thus we conclude that `gcd` for polynomials is unique up to a nonzero number multiple.

By tracing the euclidean algorithm between 102 and 45 backwards, we have

**3** = 12 - 1×9

= 12 - 1×(45 - 3×12) = (-1)×45 + (1 + 3)times;12

= (-1)×45 + 4×(102 - 2×45) = 4×102 + (-1 -8)×45

= 4×102 - 9×45.

Similarly, from the example above, we have

1 = (`t`^{2} -` t` + 1) - (`t`/3)(3`t` - 3)

= (`t`^{2} -` t` + 1) - (`t`/3)[(`t`^{4} + `t`^{3} + 3`t`^{2} + `t` + 1) - (`t`^{2} + 2`t` + 4)(`t`^{2} -` t` + 1)]

= (-`t`/3)(`t`^{4} + `t`^{3} + 3`t`^{2} + `t` + 1) + [1 + (`t`/3)(`t`^{2} + 2`t` + 4)](`t`^{2} -` t` + 1)

= [(-1/3)`t`](`t`^{4} + `t`^{3} + 3`t`^{2} + `t` + 1) + [(1/3)`t`^{3} + (2/3)`t`^{2} + (4/3)`t` + 1](`t`^{2} -` t` + 1).

Such computations can always be carried out in general, and we the following conclusion.