Question
(8) All the common fictors of 24 and 36 . (9) The lowest three common multiples of 8 and 12 . 75,225 \( 5,63,105,315 \) \( 45,55,99,165 \), factor is 45 \( \qquad \) Il it a common multiple, ommon multiples, but it tiple (LCM). Consider the numbers 54 and 648 . (1) Write both numbers as a product of primes. (2) Is 54 a factor of 648 ? (c) Consider the numbers 96 and 448 . (1) Write both numbers as a product of primes. (2) Is 448 a multiple of 96 ? (d) Determine the highest common factor (HCF) of cach of the following lists of numbers: (1) \( 12 ; 18 \) (2) \( 14 ; 15 \) (5) \( 72 ; 108 \) (6) \( 504 ; 648 \) (9) \( 72 ; 90 ; 324 \) (10) \( 225 ; 315 ; 2205 \) (7) \( 15 ; 30 ; 45 \) (4) \( 24 ; 36 ; 60 \) (8) \( 125 ; 352 \) (c) Determine the lowest common multiple (LCM) of each of the following lists of numbers: (1) \( 4 ; 6 \) (2) \( 12 ; 15 \) (3) \( 3 ; 4 ; 24 \) (4) \( 24 ; 36 \) (5) \( 72 ; 252 \) (6) \( 270 ; 300 \) (10) \( 10 ; 92 ; 115 \) (7) \( 135 ; 315 \) (8) \( 28 ; 196 ; 280 \) (f) Two very large numbers, number \( A \) and number \( B \), are decomposed into prime numbers as follows: number \( \mathrm{A}=2^{1000} \times 3^{100} \times 5^{20} \times 7^{3} \) number \( \mathrm{B}=2^{2000} \times 3^{50} \times 5^{30} \times 11 \) (1) Write down the HCF of these numbers, as a product of prime factors. (2) Write down the LCM of these numbers, as a product of prime factors. RATIOS
Ask by Hodgson Tyler. in South Africa
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Simplified Answers:**
1. **Common Factors of 24 and 36:**
- Common factors: 1, 2, 3, 4, 6, 12
2. **Lowest Three Common Multiples of 8 and 12:**
- Common multiples: 24, 48, 72
3. **Prime Factorization and Multiples:**
- **54:** \(2^1 \times 3^3\)
- **648:** \(2^3 \times 3^4\)
- **54 is a factor of 648:** Yes
4. **Prime Factorization and Multiples:**
- **96:** \(2^5 \times 3^1\)
- **448:** \(2^6 \times 7^1\)
- **448 is a multiple of 96:** No
5. **Highest Common Factor (HCF):**
- \(12\) and \(18\): 6
- \(14\) and \(15\): 1
- \(72\) and \(108\): 36
- \(504\) and \(648\): 72
- \(72\), \(90\), \(324\): 18
- \(225\), \(315\), \(2205\): 15
- \(15\), \(30\), \(45\): 15
- \(24\), \(36\), \(60\): 12
- \(125\), \(352\): 1
6. **Lowest Common Multiple (LCM):**
- \(4\) and \(6\): 12
- \(12\) and \(15\): 60
- \(3\), \(4\), \(24\): 24
- \(24\) and \(36\): 72
- \(72\) and \(252\): 504
- \(270\) and \(300\): 2700
- \(10\), \(92\), \(115\): 1150
- \(135\) and \(315\): 945
- \(28\), \(196\), \(280\): 3920
7. **HCF and LCM of Large Numbers:**
- **HCF of \(A\) and \(B\):** \(2^{1000} \times 3^{100} \times 5^{20} \times 7^{3}\)
- **LCM of \(A\) and \(B\):** \(2^{2000} \times 3^{100} \times 5^{30} \times 11\)
**Note:** The last part of the question seems incomplete.
Solution
Let's break down the problems step by step.
### Problem 8: Common Factors of 24 and 36
1. **Find the common factors of 24 and 36.**
- The factors of 24 are: \(1, 2, 3, 4, 6, 8, 12, 24\)
- The factors of 36 are: \(1, 2, 3, 4, 6, 9, 12, 18, 36\)
- The common factors are: \(1, 2, 3, 4, 6, 12\)
### Problem 9: Lowest Three Common Multiples of 8 and 12
1. **Find the least common multiple (LCM) of 8 and 12.**
- The multiples of 8 are: \(8, 16, 24, 32, 40, 48, \ldots\)
- The multiples of 12 are: \(12, 24, 36, 48, 60, \ldots\)
- The common multiples are: \(24, 48, 72, \ldots\)
- The lowest three common multiples are: \(24, 48, 72\)
### Problem 10: Prime Factorization and Multiples
#### (a) For the numbers 54 and 648:
1. **Write both numbers as a product of primes.**
- \(54 = 2^1 \times 3^3\)
- \(648 = 2^3 \times 3^4\)
2. **Is 54 a factor of 648?**
- Yes, since \(648 \div 54 = 12\), which is an integer.
#### (b) For the numbers 96 and 448:
1. **Write both numbers as a product of primes.**
- \(96 = 2^5 \times 3^1\)
- \(448 = 2^6 \times 7^1\)
2. **Is 448 a multiple of 96?**
- No, since \(448 \div 96 \approx 4.67\), which is not an integer.
### Problem 11: Highest Common Factor (HCF)
1. **HCF of \(12\) and \(18\)**:
- Factors: \(12 = 2^2 \times 3^1\), \(18 = 2^1 \times 3^2\)
- HCF: \(2^1 \times 3^1 = 6\)
2. **HCF of \(14\) and \(15\)**:
- Factors: \(14 = 2^1 \times 7^1\), \(15 = 3^1 \times 5^1\)
- HCF: \(1\)
3. **HCF of \(72\) and \(108\)**:
- Factors: \(72 = 2^3 \times 3^2\), \(108 = 2^2 \times 3^3\)
- HCF: \(2^2 \times 3^2 = 36\)
4. **HCF of \(504\) and \(648\)**:
- Factors: \(504 = 2^3 \times 3^2 \times 7^1\), \(648 = 2^3 \times 3^4\)
- HCF: \(2^3 \times 3^2 = 72\)
5. **HCF of \(72, 90, 324\)**:
- Factors: \(72 = 2^3 \times 3^2\), \(90 = 2^1 \times 3^2 \times 5^1\), \(324 = 2^2 \times 3^4\)
- HCF: \(2^1 \times 3^2 = 18\)
6. **HCF of \(225, 315, 2205\)**:
- Factors: \(225 = 3^2 \times 5^2\), \(315 = 3^2 \times 5^1 \times 7^1\), \(2205 = 3^1 \times 5^1 \times 7^2\)
- HCF: \(3^1 \times 5^1 = 15\)
7. **HCF of \(15, 30, 45\)**:
- Factors: \(15 = 3^1 \times 5^1\), \(30 = 2^1 \times 3^1 \times 5^1\), \(45 = 3^2 \times 5^1\)
- HCF: \(3^1 \times 5^1 = 15\)
8. **HCF of \(24, 36, 60\)**:
- Factors: \(24 = 2^3 \times 3^1\), \(36 = 2^2 \times 3^2\), \(60 = 2^2 \times 3^1 \times 5^1\)
- HCF: \(2^2 \times 3^1 = 12\)
9. **HCF of \(125, 352\)**:
- Factors: \(125 = 5^3\), \(352 = 2^5 \times 11^1\)
- HCF: \(1\)
### Problem 12: Lowest Common Multiple (LCM)
1. **LCM of \(4\) and \(6\)**:
- LCM: \(12\)
2. **LCM of \(12\) and \(15\)**:
- LCM: \(60\)
3. **LCM of \(3, 4, 24\)**:
- LCM: \(24\)
4. **LCM of \(24\) and \(36\)**:
- LCM: \(72\)
5. **LCM of \(72\) and \(252\)**:
- LCM: \(504\)
6. **LCM of \(270\) and \(300\)**:
- LCM: \(2700\)
7. **LCM of \(10, 92, 115\)**:
- LCM: \(1150\)
8. **LCM of \(135\) and \(315\)**:
- LCM: \(945\)
9. **LCM of \(28, 196, 280\)**:
- LCM: \(3920\)
### Problem 13: HCF and LCM of Large Numbers
1. **HCF of \(A\) and \(B\)**:
- \(A = 2^{1000} \times 3^{100} \times 5^{20} \times 7^{3}\)
-
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Simplify this solution Beyond the Answer
To find the common factors of 24 and 36, we can break them down into their prime factors. The prime factorization of 24 is \(2^3 \times 3^1\) and for 36, it’s \(2^2 \times 3^2\). The common factors come from the lowest powers of these primes, which gives us \(2^0\) (1), \(2^1\) (2), \(3^0\) (1), and \(3^1\) (3). Thus, the common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12. When looking for the lowest three common multiples (LCM) of 8 and 12, we first find the prime factorization: 8 is \(2^3\) and 12 is \(2^2 \times 3^1\). The LCM takes the highest powers of all prime factors involved, which gives us \(2^3 \times 3^1 = 24\). The first three common multiples are then 24, 48, and 72—numbers that keep showing up in the multiplication dance of both 8 and 12!
