Assertion (A) : The product of two successive positive integral multiples of 5 is 300, then the two numbers are 15 and 20.
Reason (R ) : The product of two consecutive integers is a multiple of 2

Options

A. Both A and R are true and R is the correct explanation for A.
B. Both A and R are true and R is not correct explanation for A.
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem, we will analyze the assertion and the reason provided.

### Step 1: Understanding the Assertion
The assertion states that the product of two successive positive integral multiples of 5 is 300, and these numbers are 15 and 20.

### Step 2: Define the Multiples
Let the first multiple of 5 be represented as \( 5k \), where \( k \) is a positive integer. The next successive multiple of 5 would then be \( 5(k + 1) \).

### Step 3: Set Up the Equation
According to the assertion, the product of these two multiples equals 300:
\[ (5k) \times (5(k + 1)) = 300 \]
This simplifies to:
\[ 25k(k + 1) = 300 \]

### Step 4: Simplify the Equation
Dividing both sides by 25 gives:
\[ k(k + 1) = \frac{300}{25} = 12 \]

### Step 5: Form the Quadratic Equation
Rearranging the equation leads to:
\[ k^2 + k - 12 = 0 \]

### Step 6: Factor the Quadratic
To factor the quadratic equation, we need two numbers that multiply to -12 and add to 1. The factors are 4 and -3:
\[ (k + 4)(k - 3) = 0 \]

### Step 7: Solve for \( k \)
Setting each factor to zero gives:
1. \( k + 4 = 0 \) → \( k = -4 \) (not a positive integer)
2. \( k - 3 = 0 \) → \( k = 3 \) (valid solution)

### Step 8: Find the Multiples
Substituting \( k = 3 \) back into our expressions for the multiples:
- First multiple: \( 5k = 5 \times 3 = 15 \)
- Second multiple: \( 5(k + 1) = 5 \times 4 = 20 \)

### Step 9: Verify the Assertion
Now, we check the product:
\[ 15 \times 20 = 300 \]
Thus, the assertion is correct.

### Step 10: Analyze the Reason
The reason states that the product of two consecutive integers is a multiple of 2. Let's denote two consecutive integers as \( k \) and \( k + 1 \).

### Step 11: Check the Reason
The product of these integers is:
\[ k(k + 1) \]
This product is always even because one of the two consecutive integers must be even. Therefore, the reason is also correct.

### Conclusion
Both the assertion and the reason are true. However, the reason does not directly explain the assertion. Therefore, while both statements are true, the reasoning is not the correct explanation for the assertion.

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