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The greatest common factor between $6x$ and $8x$ is
.

The greatest common factor between $14x^2$ and $10x$ is
.

The greatest common factor between $6y^2$ and $7y^2$ is
.

The greatest common factor between $12xy^2$ and $9xy$ is
.

The greatest common factor between $6x^3\text{,}$ $2x^2\text{,}$ and $8x$ is
.

Solution

Since the prime factorization of $6x$ is $\attention{2}\cdot3\cdot \attention{x}$ ...
... and $8x$ completely factors (meaning into prime factors) as $\attention{2}\cdot2\cdot2\cdot \attention{x}\text{,}$ ...
... the greatest common factor is $2x\text{.}$

Since the prime factorization of $14x^2$ is $\attention{2}\cdot7\cdot \attention{x}\cdot x$ ...
... and $10x$ completely factors as $\attention{2}\cdot5\cdot \attention{x}\text{,}$ ...
... the greatest common factor is $2x\text{.}$

Since the prime factorization of $6y^2$ is $2\cdot3\cdot \attention{y}\cdot \attention{y}$ ...
... and $7y^2$ completely factors as $7\cdot\attention{y}\cdot\attention{y}\text{,}$ ...
... the greatest common factor is $y^2\text{.}$

Since $12xy^2$ completely factors as $2\cdot2\cdot\attention{3}\cdot \attention{x}\cdot \attention{y}\cdot y$ ...
... and the prime factorization of $9xy$ is $\attention{3}\cdot3\cdot \attention{x}\cdot \attention{y}\text{,}$ ...
... the greatest common factor is $3xy\text{.}$

Since $6x^3$ completely factors as $\attention{2}\cdot3\cdot \attention{x}\cdot x\cdot x$ ...
..., $2x^2$ completely factors as $\attention{2}\cdot\attention{x}\cdot x\text{,}$ ...
... and $8x$ completely factors as $\attention{2}\cdot2\cdot2\cdot\attention{x}\text{,}$ ...
... the greatest common factor is $2x\text{.}$