To find the L.C.M. (Least Common Multiple) of the given numbers using the prime factorization method, we will follow these steps:
### (i) Finding the L.C.M. of 27 and 90
**Step 1: Prime Factorization of 27**
- 27 can be expressed as:
- \( 27 = 3 \times 9 \)
- \( 9 = 3 \times 3 \)
- Therefore, the prime factorization of 27 is:
- \( 27 = 3^3 \)
**Step 2: Prime Factorization of 90**
- 90 can be expressed as:
- \( 90 = 9 \times 10 \)
- \( 9 = 3 \times 3 \) and \( 10 = 2 \times 5 \)
- Therefore, the prime factorization of 90 is:
- \( 90 = 2^1 \times 3^2 \times 5^1 \)
**Step 3: Write down the prime factors**
- For L.C.M., we take the highest power of each prime factor from both numbers:
- From 27: \( 3^3 \)
- From 90: \( 2^1, 3^2, 5^1 \)
**Step 4: L.C.M. Calculation**
- L.C.M. = \( 2^1 \times 3^3 \times 5^1 \)
- Now calculate:
- \( 2^1 = 2 \)
- \( 3^3 = 27 \)
- \( 5^1 = 5 \)
- Therefore:
- \( L.C.M. = 2 \times 27 \times 5 \)
- \( = 54 \times 5 \)
- \( = 270 \)
### (ii) Finding the L.C.M. of 36, 48, and 210
**Step 1: Prime Factorization of 36**
- 36 can be expressed as:
- \( 36 = 6 \times 6 \)
- \( 6 = 2 \times 3 \)
- Therefore, the prime factorization of 36 is:
- \( 36 = 2^2 \times 3^2 \)
**Step 2: Prime Factorization of 48**
- 48 can be expressed as:
- \( 48 = 16 \times 3 \)
- \( 16 = 2^4 \)
- Therefore, the prime factorization of 48 is:
- \( 48 = 2^4 \times 3^1 \)
**Step 3: Prime Factorization of 210**
- 210 can be expressed as:
- \( 210 = 2 \times 105 \)
- \( 105 = 3 \times 35 \)
- \( 35 = 5 \times 7 \)
- Therefore, the prime factorization of 210 is:
- \( 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \)
**Step 4: Write down the prime factors**
- For L.C.M., we take the highest power of each prime factor from all three numbers:
- From 36: \( 2^2, 3^2 \)
- From 48: \( 2^4, 3^1 \)
- From 210: \( 2^1, 3^1, 5^1, 7^1 \)
**Step 5: L.C.M. Calculation**
- L.C.M. = \( 2^4 \times 3^2 \times 5^1 \times 7^1 \)
- Now calculate:
- \( 2^4 = 16 \)
- \( 3^2 = 9 \)
- \( 5^1 = 5 \)
- \( 7^1 = 7 \)
- Therefore:
- \( L.C.M. = 16 \times 9 \times 5 \times 7 \)
- First calculate \( 16 \times 9 = 144 \)
- Then \( 144 \times 5 = 720 \)
- Finally \( 720 \times 7 = 5040 \)
### Final Answers:
- L.C.M. of 27 and 90 is **270**
- L.C.M. of 36, 48, and 210 is **5040**
