To find the least common multiple (LCM) of the numbers 102, 119, and 153 using the division method, we first need to identify the prime factors of each number.
1. **Prime factorization**:
- For 102: 102 can be divided by 2 (the smallest prime number):
- 102 ÷ 2 = 51
- Next, 51 can be divided by 3:
- 51 ÷ 3 = 17
- 17 is a prime number, so the prime factorization of 102 is:
- 102 = 2 × 3 × 17
- For 119: 119 can be divided by 7:
- 119 ÷ 7 = 17
- 17 is a prime number, so the prime factorization of 119 is:
- 119 = 7 × 17
- For 153: 153 can be divided by 3:
- 153 ÷ 3 = 51
- Then, we already know from above that 51 can be factored further:
- 51 = 3 × 17
- Therefore, the prime factorization of 153 is:
- 153 = 3 × 3 × 17 = 3² × 17
2. **List of prime factors**:
Now we have the following prime factors:
- 102 = 2 × 3 × 17
- 119 = 7 × 17
- 153 = 3² × 17
3. **Using the division method**:
We will now set up a division table using all the prime factors. The LCM is found by multiplying the highest power of each prime number that appears in the factorization of all three numbers:
| Divisor | Remainder |
|---|---|
| 2 | 51, 119, 153 |
| 3 | 17, 119, 51 |
| 7 | 17, 17, 51 |
| 17 | 1, 1, 3 |
4. **Calculating the LCM**:
Now we multiply all the prime factors we used:
- LCM = 21 × 32 × 71 × 171
Calculating this gives:
- LCM = 2 × 9 × 7 × 17
- LCM = 126 × 17 = 2142
Thus, the LCM of 102, 119, and 153 is 2142.