Let's simplify the expression step-by-step:
The expression is:
\[ 4 \div 4 \cdot 2 \cdot 2 \div (15 - 13) \]
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Start by simplifying the parentheses: \[ 15 - 13 = 2 \]
Now, substitute back into the expression: \[ 4 \div 4 \cdot 2 \cdot 2 \div 2 \]
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Now, perform the division and multiplication from left to right:
First, calculate \(4 \div 4\): \[ 4 \div 4 = 1 \]
Now our expression is: \[ 1 \cdot 2 \cdot 2 \div 2 \]
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Next, calculate \(1 \cdot 2\): \[ 1 \cdot 2 = 2 \]
So now we have: \[ 2 \cdot 2 \div 2 \]
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Next, calculate \(2 \cdot 2\): \[ 2 \cdot 2 = 4 \]
Now our expression is: \[ 4 \div 2 \]
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Finally, calculate \(4 \div 2\): \[ 4 \div 2 = 2 \]
Thus, the fully simplified expression is:
\[ \boxed{2} \]