Find the identity element for the binary operation * on Z defined as a*b = a + b-5. Also find the inverse element of 100.

Answer

To solve the problem, we will find the identity element for the binary operation defined on \( \mathbb{Z} \) as \( a * b = a + b - 5 \), and then we will find the inverse element of 100.

### Step 1: Find the Identity Element

1. **Definition of Identity Element**: An element \( e \) is called the identity element if for every element \( a \) in the set, the following holds:
\[ a * e = a \quad \text{and} \quad e * a = a \]

2. **Using the Binary Operation**: According to the given operation, we have:
\[ a * e = a + e - 5 \]

3. **Setting up the Equation**: For \( e \) to be the identity element, we need:
\[ a + e - 5 = a \]

4. **Simplifying the Equation**: We can cancel \( a \) from both sides:
\[ e - 5 = 0 \]

5. **Solving for \( e \)**: Adding 5 to both sides gives:
\[ e = 5 \]

Thus, the identity element is **5**.

### Step 2: Find the Inverse Element of 100

1. **Definition of Inverse Element**: An element \( b \) is called the inverse of \( a \) if:
\[ a * b = e \]
where \( e \) is the identity element we found.

2. **Using the Binary Operation**: For \( a = 100 \) and \( e = 5 \):
\[ 100 * b = 5 \]

3. **Setting up the Equation**: According to the operation:
\[ 100 + b - 5 = 5 \]

4. **Simplifying the Equation**: Rearranging gives:
\[ 100 + b - 5 = 5 \implies 100 + b = 10 \]

5. **Solving for \( b \)**: Subtracting 100 from both sides:
\[ b = 10 - 100 = -90 \]

Thus, the inverse element of 100 is **-90**.

### Final Answers:
- The identity element is **5**.
- The inverse of 100 is **-90**.