(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 $$ \mathrm{A}=2^{1000} imes 3^{100} imes 5^{20} imes 7^{3} $$ number $$ \mathrm{B}=2^{2000} imes 3^{50} imes 5^{30} imes 11 $$ (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

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 $$ \mathrm{A}=2^{1000} 	imes 3^{100} 	imes 5^{20} 	imes 7^{3} $$ number $$ \mathrm{B}=2^{2000} 	imes 3^{50} 	imes 5^{30} 	imes 11 $$ (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

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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.**
   - $$54 = 2^1 	imes 3^3$$
   - $$648 = 2^3 	imes 3^4$$

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.**
   - $$96 = 2^5 	imes 3^1$$
   - $$448 = 2^6 	imes 7^1$$

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: $$12 = 2^2 	imes 3^1$$, $$18 = 2^1 	imes 3^2$$
   - HCF: $$2^1 	imes 3^1 = 6$$

2. **HCF of $$14$$ and $$15$$**:
   - Factors: $$14 = 2^1 	imes 7^1$$, $$15 = 3^1 	imes 5^1$$
   - HCF: $$1$$

3. **HCF of $$72$$ and $$108$$**:
   - Factors: $$72 = 2^3 	imes 3^2$$, $$108 = 2^2 	imes 3^3$$
   - HCF: $$2^2 	imes 3^2 = 36$$

4. **HCF of $$504$$ and $$648$$**:
   - Factors: $$504 = 2^3 	imes 3^2 	imes 7^1$$, $$648 = 2^3 	imes 3^4$$
   - HCF: $$2^3 	imes 3^2 = 72$$

5. **HCF of $$72, 90, 324$$**:
   - Factors: $$72 = 2^3 	imes 3^2$$, $$90 = 2^1 	imes 3^2 	imes 5^1$$, $$324 = 2^2 	imes 3^4$$
   - HCF: $$2^1 	imes 3^2 = 18$$

6. **HCF of $$225, 315, 2205$$**:
   - Factors: $$225 = 3^2 	imes 5^2$$, $$315 = 3^2 	imes 5^1 	imes 7^1$$, $$2205 = 3^1 	imes 5^1 	imes 7^2$$
   - HCF: $$3^1 	imes 5^1 = 15$$

7. **HCF of $$15, 30, 45$$**:
   - Factors: $$15 = 3^1 	imes 5^1$$, $$30 = 2^1 	imes 3^1 	imes 5^1$$, $$45 = 3^2 	imes 5^1$$
   - HCF: $$3^1 	imes 5^1 = 15$$

8. **HCF of $$24, 36, 60$$**:
   - Factors: $$24 = 2^3 	imes 3^1$$, $$36 = 2^2 	imes 3^2$$, $$60 = 2^2 	imes 3^1 	imes 5^1$$
   - HCF: $$2^2 	imes 3^1 = 12$$

9. **HCF of $$125, 352$$**:
   - Factors: $$125 = 5^3$$, $$352 = 2^5 	imes 11^1$$
   - 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 $$B$$**:
   - $$A = 2^{1000} 	imes 3^{100} 	imes 5^{20} 	imes 7^{3}$$
   -
**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:** $$2^1 	imes 3^3$$ - **648:** $$2^3 	imes 3^4$$ - **54 is a factor of 648:** Yes 4. **Prime Factorization and Multiples:** - **96:** $$2^5 	imes 3^1$$ - **448:** $$2^6 	imes 7^1$$ - **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 $$B$$:** $$2^{1000} 	imes 3^{100} 	imes 5^{20} 	imes 7^{3}$$ - **LCM of $$A$$ and $$B$$:** $$2^{2000} 	imes 3^{100} 	imes 5^{30} 	imes 11$$ **Note:** The last part of the question seems incomplete.

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