To find the greatest common factor (GCF) of the expressions \( 22axy \) and \( 11bx \), we will first factor each expression.
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Factor \( 22axy \):
- The number 22 can be factored into \( 2 \times 11 \).
- Therefore, the expression \( 22axy \) can be written as: \[ 22axy = 2 \times 11 \times a \times x \times y \]
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Factor \( 11bx \):
- The number 11 is prime, thus its only factors are 1 and 11.
- Therefore, the expression \( 11bx \) can be written as: \[ 11bx = 11 \times b \times x \]
Now we will identify the GCF by looking for common factors in both expressions:
- The common numerical factor between \( 22axy \) and \( 11bx \) is 11.
- The common variable factor between the two expressions is \( x \), as \( x \) appears in both.
From this, the GCF can be computed as the product of the common numerical and variable factors:
\[ \text{GCF} = 11 \times x \]
Thus, the greatest common factor of \( 22axy \) and \( 11bx \) is:
\[ \boxed{11x} \]