What are the common factors of 18xy and 32xy?

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Answer 1 · 2016-12-08T08:32:58

The common factors with integer coefficients are:

$\pm 1, \pm 2, \pm x, \pm 2 x, \pm y, \pm 2 y, \pm x y, \pm 2 x y$ $+-1, +-2, +-x, +-2x, +-y, +-2y, +-xy, +-2xy$

The prime factorisations of $18$ $18$ and $32$ $32$ are:

$18 = 2 \cdot 3^{2}$ $18 = 2*3^2$

$32 = 2^{5}$ $32 = 2^5$

Hence their common positive integer factors are:

$1, 2$ $1, 2$

If you include negative integer factors then they have common factors:

$\pm 1, \pm 2$ $+-1, +-2$

Both $18 x y$ $18xy$ and $32 x y$ $32xy$ are multiples of $x$ $x$ and $y$ $y$ , so ignoring scalar multipliers, their common polynomial factors are:

$1, x, y, x y$ $1, x, y, xy$

Putting all of the possible combinations together, all of their polynomial factors with integer coefficients are:

$\pm 1, \pm 2, \pm x, \pm 2 x, \pm y, \pm 2 y, \pm x y, \pm 2 x y$ $+-1, +-2, +-x, +-2x, +-y, +-2y, +-xy, +-2xy$

The greatest common factor of $18 x y$ $18xy$ and $32 x y$ $32xy$ is $2 x y$ $2xy$