The relation between H.C.F. and L.C.M of 12 and 20 will be

Options

A. H.C.F gt L. C. M
B. H.C.F lt L. C. M
C. H.C.F = L. C. M
D. None of these

Answer

To find the relation between the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of the numbers 12 and 20, we will follow these steps:

### Step 1: Find the L.C.M. of 12 and 20

To find the L.C.M., we can use the prime factorization method.

1. **Prime Factorization of 12:**
- 12 = 2 × 2 × 3 = \(2^2 \times 3^1\)

2. **Prime Factorization of 20:**
- 20 = 2 × 2 × 5 = \(2^2 \times 5^1\)

3. **Finding the L.C.M.:**
- Take the highest power of each prime factor:
- For 2: \(2^2\)
- For 3: \(3^1\)
- For 5: \(5^1\)

Therefore,
\[ \text{L.C.M.} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Find the H.C.F. of 12 and 20

To find the H.C.F., we can also use the prime factorization method.

1. **Using the prime factorizations:**
- For 12: \(2^2 \times 3^1\)
- For 20: \(2^2 \times 5^1\)

2. **Finding the H.C.F.:**
- Take the lowest power of each common prime factor:
- For 2: \(2^2\) (common factor)
- For 3: Not common
- For 5: Not common

Therefore,
\[ \text{H.C.F.} = 2^2 = 4 \]

### Step 3: Establish the Relation between H.C.F. and L.C.M.

Now we have:
- H.C.F. = 4
- L.C.M. = 60

We can observe that:
\[ \text{H.C.F.} < \text{L.C.M.} \]

### Conclusion

The relation between the H.C.F. and L.C.M. of 12 and 20 is that the H.C.F. is less than the L.C.M.

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