Can two numbers have 15 as their H.C.F. and 110 as their L.C.M.? Give reason to justify your answer.

Answer

To determine whether two numbers can have 15 as their H.C.F. (Highest Common Factor) and 110 as their L.C.M. (Lowest Common Multiple), we can follow these steps:

### Step 1: Understand the relationship between H.C.F. and L.C.M.
The relationship between H.C.F. and L.C.M. of two numbers can be expressed by the formula:
\[ \text{H.C.F.} \times \text{L.C.M.} = \text{Product of the two numbers} \]
This means that the product of the H.C.F. and L.C.M. should equal the product of the two numbers.

### Step 2: Substitute the given values into the formula.
Given:
- H.C.F. = 15
- L.C.M. = 110

Using the formula:
\[ 15 \times 110 = \text{Product of the two numbers} \]
Calculating this gives:
\[ 15 \times 110 = 1650 \]

### Step 3: Check if L.C.M. is divisible by H.C.F.
For two numbers to have 15 as their H.C.F. and 110 as their L.C.M., the L.C.M. must be divisible by the H.C.F. This means:
\[ \frac{110}{15} \]
Calculating this gives:
\[ 110 \div 15 = 7.33 \quad (\text{not an integer}) \]
Since 110 is not divisible by 15, this indicates that the condition is not satisfied.

### Step 4: Conclusion
Since the L.C.M. (110) is not divisible by the H.C.F. (15), it is not possible for two numbers to have 15 as their H.C.F. and 110 as their L.C.M.

### Final Answer
No, two numbers cannot have 15 as their H.C.F. and 110 as their L.C.M. because 110 is not divisible by 15.

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