Write the negative of the proposition : "If a number is divisible by 15, then it is divisible by 5 or 3".

Answer

To find the negative of the proposition "If a number is divisible by 15, then it is divisible by 5 or 3", we will follow these steps:

### Step 1: Identify the Statements
Let:
- \( p \): A number is divisible by 15.
- \( q \): A number is divisible by 5.
- \( r \): A number is divisible by 3.

The original proposition can be expressed as:
\[ p \implies (q \lor r) \]
This means "If \( p \) is true, then \( q \) or \( r \) is true."

### Step 2: Write the Negation of the Proposition
The negation of the implication \( p \implies (q \lor r) \) can be expressed using the property:
\[ \neg (p \implies (q \lor r)) \equiv p \land \neg (q \lor r) \]

### Step 3: Apply De Morgan's Law
Now we need to negate \( (q \lor r) \):
Using De Morgan's Law, we have:
\[ \neg (q \lor r) \equiv \neg q \land \neg r \]

### Step 4: Combine the Results
Thus, the negation of the original proposition becomes:
\[ p \land (\neg q \land \neg r) \]
This can be rewritten as:
\[ p \land \neg q \land \neg r \]

### Step 5: Interpret the Result
Now, substituting back the meanings of \( p \), \( q \), and \( r \):
- \( p \): A number is divisible by 15.
- \( \neg q \): A number is not divisible by 5.
- \( \neg r \): A number is not divisible by 3.

So, the final negation of the proposition is:
"A number is divisible by 15 and is not divisible by 5 and not divisible by 3."

### Summary of the Solution
The negative of the proposition "If a number is divisible by 15, then it is divisible by 5 or 3" is:
"A number is divisible by 15 and is not divisible by 5 and not divisible by 3."

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