To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 150 and 240 using prime factorization, we will follow these steps:
### Step 1: Prime Factorization of 150
To find the prime factorization of 150, we divide it by the smallest prime numbers until we reach 1.
1. Divide by 2:
\( 150 \div 2 = 75 \)
So, we have \( 150 = 2 \times 75 \).
2. Divide by 3:
\( 75 \div 3 = 25 \)
So, \( 75 = 3 \times 25 \).
Thus, \( 150 = 2 \times 3 \times 25 \).
3. Divide by 5:
\( 25 \div 5 = 5 \)
So, \( 25 = 5 \times 5 \).
Thus, \( 150 = 2 \times 3 \times 5 \times 5 \).
The prime factorization of 150 is:
\[ 150 = 2^1 \times 3^1 \times 5^2 \]
### Step 2: Prime Factorization of 240
Next, we perform the same process for 240.
1. Divide by 2:
\( 240 \div 2 = 120 \)
So, \( 240 = 2 \times 120 \).
2. Divide by 2 again:
\( 120 \div 2 = 60 \)
So, \( 120 = 2 \times 60 \).
Thus, \( 240 = 2^2 \times 60 \).
3. Divide by 2 again:
\( 60 \div 2 = 30 \)
So, \( 60 = 2 \times 30 \).
Thus, \( 240 = 2^3 \times 30 \).
4. Divide by 2 again:
\( 30 \div 2 = 15 \)
So, \( 30 = 2 \times 15 \).
Thus, \( 240 = 2^4 \times 15 \).
5. Divide by 3:
\( 15 \div 3 = 5 \)
So, \( 15 = 3 \times 5 \).
Thus, \( 240 = 2^4 \times 3^1 \times 5^1 \).
The prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]
### Step 3: Finding HCF
To find the HCF, we take the lowest power of each common prime factor from both factorizations.
- For \( 2 \): The minimum power is \( 2^1 \).
- For \( 3 \): The minimum power is \( 3^1 \).
- For \( 5 \): The minimum power is \( 5^1 \).
Thus, the HCF is:
\[ HCF = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 \]
### Step 4: Finding LCM
To find the LCM, we take the highest power of each prime factor from both factorizations.
- For \( 2 \): The maximum power is \( 2^4 \).
- For \( 3 \): The maximum power is \( 3^1 \).
- For \( 5 \): The maximum power is \( 5^2 \).
Thus, the LCM is:
\[ LCM = 2^4 \times 3^1 \times 5^2 = 16 \times 3 \times 25 \]
Calculating this step by step:
1. \( 16 \times 3 = 48 \)
2. \( 48 \times 25 = 1200 \)
So, the LCM is:
\[ LCM = 1200 \]
### Final Answers
- HCF of 150 and 240 is **30**.
- LCM of 150 and 240 is **1200**.
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