Two numbers have 16 as their HCF and 146 as their LCM, then one can say that :

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to analyze the relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers.

1. **Explanation for Option 3 (Correct Answer):**
   - The relationship between two numbers $a$ and $b$, their HCF, and their LCM is given by the formula:
     $$
     \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b
     $$
   - Given that the HCF is 16 and the LCM is 146, we substitute these values into the formula:
     $$
     16 \times 146 = a \times b
     $$
   - Calculating the product, we get:
     $$
     2336 = a \times b
     $$
   - For two numbers to have an HCF of 16, each number must be a multiple of 16. Let $a = 16m$ and $b = 16n$, where $m$ and $n$ are coprime (since the HCF of $a$ and $b$ is 16).
   - Substituting into the equation:
     $$
     16m \times 16n = 2336
     $$
     $$
     256mn = 2336
     $$
     $$
     mn = \frac{2336}{256} = 9.125
     $$
   - Since $m$ and $n$ must be integers, and 9.125 is not an integer, no such pair of integers $m$ and $n$ exists. Therefore, no such pair of numbers $a$ and $b$ exists that satisfy the given conditions.

2. **Explanation for why other options are incorrect:**
   - **Option 1 (Many such pairs of numbers exist):** This option is incorrect because, as shown above, the equation $mn = 9.125$ does not yield integer solutions for $m$ and $n$. Therefore, multiple pairs cannot exist.
   - **Option 2 (Only one such pair of numbers exists):** This option is incorrect because the equation $mn = 9.125$ does not yield even a single integer solution for $m$ and $n$. Hence, there is not even one pair that satisfies the conditions.
   - **Option 4 (Only two such pairs of numbers exist):** This option is incorrect for the same reason as Option 1 and 2. Since $mn = 9.125$ does not provide integer solutions, there cannot be two pairs of numbers that satisfy the conditions.

In conclusion, Option 3 is the correct answer because the mathematical conditions required for the existence of such a pair of numbers are not met.