Find the H.C.F. of 180 and 336. Hence, find their L.C.M.

Answer

To find the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of the numbers 180 and 336, we will follow these steps:

### Step 1: Prime Factorization of 180
1. Divide 180 by 2:
\( 180 \div 2 = 90 \)
2. Divide 90 by 2:
\( 90 \div 2 = 45 \)
3. Divide 45 by 3:
\( 45 \div 3 = 15 \)
4. Divide 15 by 3:
\( 15 \div 3 = 5 \)
5. Finally, divide 5 by 5:
\( 5 \div 5 = 1 \)

So, the prime factorization of 180 is:
\[ 180 = 2^2 \times 3^2 \times 5^1 \]

### Step 2: Prime Factorization of 336
1. Divide 336 by 2:
\( 336 \div 2 = 168 \)
2. Divide 168 by 2:
\( 168 \div 2 = 84 \)
3. Divide 84 by 2:
\( 84 \div 2 = 42 \)
4. Divide 42 by 2:
\( 42 \div 2 = 21 \)
5. Divide 21 by 3:
\( 21 \div 3 = 7 \)
6. Finally, divide 7 by 7:
\( 7 \div 7 = 1 \)

So, the prime factorization of 336 is:
\[ 336 = 2^4 \times 3^1 \times 7^1 \]

### Step 3: Finding H.C.F.
To find the H.C.F., we take the lowest power of all common prime factors:
- For 2: The minimum power is \( 2^2 \) (from 180).
- For 3: The minimum power is \( 3^1 \) (from 336).

Now, multiply these together:
\[ \text{H.C.F.} = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Step 4: Finding L.C.M.
To find the L.C.M., we take the highest power of all prime factors present in either number:
- For 2: The maximum power is \( 2^4 \) (from 336).
- For 3: The maximum power is \( 3^2 \) (from 180).
- For 5: The maximum power is \( 5^1 \) (from 180).
- For 7: The maximum power is \( 7^1 \) (from 336).

Now, multiply these together:
\[ \text{L.C.M.} = 2^4 \times 3^2 \times 5^1 \times 7^1 \]
Calculating this step-by-step:
1. \( 2^4 = 16 \)
2. \( 3^2 = 9 \)
3. \( 5^1 = 5 \)
4. \( 7^1 = 7 \)

Now multiply these results:
\[ 16 \times 9 = 144 \]
\[ 144 \times 5 = 720 \]
\[ 720 \times 7 = 5040 \]

Thus, the L.C.M. is:
\[ \text{L.C.M.} = 5040 \]

### Final Answer:
- H.C.F. of 180 and 336 is **12**.
- L.C.M. of 180 and 336 is **5040**.

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