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If a*b = 10+ a + b for all `a,b in Z` find the identity element and inverse of 25.

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To solve the problem, we will follow these steps: ### Step 1: Find the Identity Element We need to find the identity element \( e \) such that for any integer \( a \): \[ a * e = a \] Given the operation defined as \( a * b = 10 + a + b \), we can substitute \( e \) into the equation: \[ a * e = 10 + a + e \] Setting this equal to \( a \): \[ 10 + a + e = a \] Now, we can simplify this equation by subtracting \( a \) from both sides: \[ 10 + e = 0 \] To isolate \( e \), we subtract 10 from both sides: \[ e = -10 \] Thus, the identity element is \( e = -10 \). ### Step 2: Find the Inverse of 25 Next, we need to find the inverse of \( 25 \). Let \( b \) be the inverse of \( 25 \). By definition, the inverse satisfies: \[ 25 * b = e \] Substituting \( e = -10 \): \[ 25 * b = -10 \] Using the operation definition: \[ 10 + 25 + b = -10 \] Now, we can simplify this equation: \[ 35 + b = -10 \] To find \( b \), we subtract 35 from both sides: \[ b = -10 - 35 \] This simplifies to: \[ b = -45 \] Thus, the inverse of \( 25 \) is \( -45 \). ### Final Answers - The identity element is \( -10 \). - The inverse of \( 25 \) is \( -45 \). ---
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