Question
4. Find the least common multiple.12x and 33x4132x5132x43x396x4
Answers
Answered by
GPT-4o mini
To find the least common multiple (LCM) of the given expressions, we first need to break down each term into its prime factorization and identify the highest powers of all prime factors present.
### Given expressions:
1. \( 12x \)
2. \( 33x \)
3. \( 4132x \)
4. \( 5132x \)
5. \( 43x \)
6. \( 396x \)
7. \( 64 \)
### Factorization:
1. **\( 12x \)**:
\( 12 = 2^2 \cdot 3 \)
This gives us \( 12x = 2^2 \cdot 3 \cdot x \).
2. **\( 33x \)**:
\( 33 = 3 \cdot 11 \)
This gives us \( 33x = 3 \cdot 11 \cdot x \).
3. **\( 4132x \)**:
\( 4132 = 2^2 \cdot 1033 \) (since 1033 is a prime number)
This gives us \( 4132x = 2^2 \cdot 1033 \cdot x \).
4. **\( 5132x \)**:
\( 5132 = 2^2 \cdot 1283 \) (since 1283 is a prime number)
This gives us \( 5132x = 2^2 \cdot 1283 \cdot x \).
5. **\( 43x \)**:
\( 43 \) is a prime number, thus
This gives us \( 43x = 43 \cdot x \).
6. **\( 396x \)**:
\( 396 = 2^2 \cdot 3^2 \cdot 11 \)
This gives us \( 396x = 2^2 \cdot 3^2 \cdot 11 \cdot x \).
7. **\( 64 \)**:
\( 64 = 2^6 \)
This gives us \( 64 = 2^6 \).
### Finding the LCM:
The LCM is found by taking the highest power of each prime factor from the above factorizations:
- For \( 2 \): The highest power is \( 2^6 \) (from \( 64 \)).
- For \( 3 \): The highest power is \( 3^2 \) (from \( 396x \)).
- For \( 11 \): The highest power is \( 11^1 \) (from \( 33x \) and \( 396x \)).
- For \( 43 \): The highest power is \( 43^1 \) (from \( 43x \)).
- For \( 1033 \): The highest power is \( 1033^1 \) (from \( 4132x \)).
- For \( 1283 \): The highest power is \( 1283^1 \) (from \( 5132x \)).
### LCM Calculation:
\[
\text{LCM} = 2^6 \cdot 3^2 \cdot 11^1 \cdot 43^1 \cdot 1033^1 \cdot 1283^1 \cdot x
\]
You can leave the LCM in this product form, or calculate numerically if needed, but the expression itself gives a complete answer in terms of the LCM of these algebraic expressions.
### Given expressions:
1. \( 12x \)
2. \( 33x \)
3. \( 4132x \)
4. \( 5132x \)
5. \( 43x \)
6. \( 396x \)
7. \( 64 \)
### Factorization:
1. **\( 12x \)**:
\( 12 = 2^2 \cdot 3 \)
This gives us \( 12x = 2^2 \cdot 3 \cdot x \).
2. **\( 33x \)**:
\( 33 = 3 \cdot 11 \)
This gives us \( 33x = 3 \cdot 11 \cdot x \).
3. **\( 4132x \)**:
\( 4132 = 2^2 \cdot 1033 \) (since 1033 is a prime number)
This gives us \( 4132x = 2^2 \cdot 1033 \cdot x \).
4. **\( 5132x \)**:
\( 5132 = 2^2 \cdot 1283 \) (since 1283 is a prime number)
This gives us \( 5132x = 2^2 \cdot 1283 \cdot x \).
5. **\( 43x \)**:
\( 43 \) is a prime number, thus
This gives us \( 43x = 43 \cdot x \).
6. **\( 396x \)**:
\( 396 = 2^2 \cdot 3^2 \cdot 11 \)
This gives us \( 396x = 2^2 \cdot 3^2 \cdot 11 \cdot x \).
7. **\( 64 \)**:
\( 64 = 2^6 \)
This gives us \( 64 = 2^6 \).
### Finding the LCM:
The LCM is found by taking the highest power of each prime factor from the above factorizations:
- For \( 2 \): The highest power is \( 2^6 \) (from \( 64 \)).
- For \( 3 \): The highest power is \( 3^2 \) (from \( 396x \)).
- For \( 11 \): The highest power is \( 11^1 \) (from \( 33x \) and \( 396x \)).
- For \( 43 \): The highest power is \( 43^1 \) (from \( 43x \)).
- For \( 1033 \): The highest power is \( 1033^1 \) (from \( 4132x \)).
- For \( 1283 \): The highest power is \( 1283^1 \) (from \( 5132x \)).
### LCM Calculation:
\[
\text{LCM} = 2^6 \cdot 3^2 \cdot 11^1 \cdot 43^1 \cdot 1033^1 \cdot 1283^1 \cdot x
\]
You can leave the LCM in this product form, or calculate numerically if needed, but the expression itself gives a complete answer in terms of the LCM of these algebraic expressions.