CISE 1.5 Fnd the HCF for these sets of numbers. b) 108 and 270 21 and 84 d) 195 and 330 30,210 and 700 Find the LCM for these sets of numbers. \begin{tabular}{ll} a) & b) \ \hline \end{tabular} $$ \begin{array}{ll} ext { a) } 24 ext { and } 90 & ext { b) } \ ext { and } 48\end{array} $$ Mr Abrahams is able to divide the learners in his Mathematics class into ec groups of 2,5 or 6 without leaving any learners out. What is the smallest possible number of learners in the class? bell rings every 30 minutes at the Junior School and eve possibll rings every 30 minools start lessons at $$ 8: 00 $$ a.m.

Question

CISE 1.5 Fnd the HCF for these sets of numbers. b) 108 and 270 21 and 84 d) 195 and 330 30,210 and 700 Find the LCM for these sets of numbers. \begin{tabular}{ll} a) & b) \ \hline \end{tabular} $$ \begin{array}{ll}	ext { a) } 24 	ext { and } 90 & 	ext { b) } \ 	ext { and } 48\end{array} $$ Mr Abrahams is able to divide the learners in his Mathematics class into ec groups of 2,5 or 6 without leaving any learners out. What is the smallest possible number of learners in the class? bell rings every 30 minutes at the Junior School and eve possibll rings every 30 minools start lessons at $$ 8: 00 $$ a.m.

UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### Part 1: Finding the HCF

**a) HCF of 108 and 270**

1. **Prime Factorization**:
   - $$ 108 = 2^2 \times 3^3 $$
   - $$ 270 = 2^1 \times 3^3 \times 5^1 $$

2. **Identify Common Factors**:
   - The common prime factors are $$ 2 $$ and $$ 3 $$.
   - The minimum power of $$ 2 $$ is $$ 2^1 $$ and for $$ 3 $$ is $$ 3^3 $$.

3. **Calculate HCF**:
   $$
   \text{HCF} = 2^1 \times 3^3 = 2 \times 27 = 54
   $$

**b) HCF of 21 and 84**

1. **Prime Factorization**:
   - $$ 21 = 3^1 \times 7^1 $$
   - $$ 84 = 2^2 \times 3^1 \times 7^1 $$

2. **Identify Common Factors**:
   - The common prime factors are $$ 3 $$ and $$ 7 $$.
   - The minimum power of $$ 3 $$ is $$ 3^1 $$ and for $$ 7 $$ is $$ 7^1 $$.

3. **Calculate HCF**:
   $$
   \text{HCF} = 3^1 \times 7^1 = 3 \times 7 = 21
   $$

**c) HCF of 195 and 330**

1. **Prime Factorization**:
   - $$ 195 = 3^1 \times 5^1 \times 13^1 $$
   - $$ 330 = 2^1 \times 3^1 \times 5^1 \times 11^1 $$

2. **Identify Common Factors**:
   - The common prime factors are $$ 3 $$ and $$ 5 $$.
   - The minimum power of $$ 3 $$ is $$ 3^1 $$ and for $$ 5 $$ is $$ 5^1 $$.

3. **Calculate HCF**:
   $$
   \text{HCF} = 3^1 \times 5^1 = 3 \times 5 = 15
   $$

**d) HCF of 30, 210, and 700**

1. **Prime Factorization**:
   - $$ 30 = 2^1 \times 3^1 \times 5^1 $$
   - $$ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 $$
   - $$ 700 = 2^2 \times 5^2 \times 7^1 $$

2. **Identify Common Factors**:
   - The common prime factors are $$ 2 $$, $$ 3 $$, and $$ 5 $$.
   - The minimum power of $$ 2 $$ is $$ 2^1 $$, for $$ 3 $$ is $$ 3^0 $$ (not present in 700), and for $$ 5 $$ is $$ 5^1 $$.

3. **Calculate HCF**:
   $$
   \text{HCF} = 2^1 \times 5^1 = 2 \times 5 = 10
   $$

### Part 2: Finding the LCM

**a) LCM of 24 and 90**

1. **Prime Factorization**:
   - $$ 24 = 2^3 \times 3^1 $$
   - $$ 90 = 2^1 \times 3^2 \times 5^1 $$

2. **Identify All Factors**:
   - The maximum power of $$ 2 $$ is $$ 2^3 $$, for $$ 3 $$ is $$ 3^2 $$, and for $$ 5 $$ is $$ 5^1 $$.

3. **Calculate LCM**:
   $$
   \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360
   $$

**b) LCM of 48 and 48**

1. **Prime Factorization**:
   - $$ 48 = 2^4 \times 3^1 $$

2. **Identify All Factors**:
   - The maximum power of $$ 2 $$ is $$ 2^4 $$ and for $$ 3 $$ is $$ 3^1 $$.

3. **Calculate LCM**:
   $$
   \text{LCM} = 2^4 \times 3^1 = 16 \times 3 = 48
   $$

### Part 3: Finding the Smallest Number of Learners

Mr. Abrahams can divide the learners into groups of 2, 5, or 6. We need to find the least common multiple (LCM) of these numbers.

1. **LCM of 2, 5, and 6**:
   - Prime Factorization:
     - $$ 2 = 2^1 $$
     - $$ 5 = 5^1 $$
     - $$ 6 = 2^1 \times 3^1 $$

2. **Identify All Factors**:
   - The maximum power of $$ 2 $$ is $$ 2^1 $$, for $$ 3 $$ is $$ 3^1 $$, and for $$ 5 $$ is $$ 5^1 $$.

3. **Calculate LCM**:
   $$
   \text{LCM} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30
   $$

Thus, the smallest possible number of learners in the class is **30**.

### Part 4: Bell Rings

The bell rings every 30 minutes starting at 8:00 a.m. The next bell will ring at 8:30 a.m., then 9:00 a.m., and so on.

If you need further calculations or clarifications, feel free to ask!
**HCF and LCM Calculations:** - **HCF of 108 and 270:** 54 - **HCF of 21 and 84:** 21 - **HCF of 195 and 330:** 15 - **HCF of 30, 210, and 700:** 10 - **LCM of 24 and 90:** 360 - **LCM of 48 and 48:** 48 **Smallest Number of Learners:** 30 **Bell Ringing Schedule:** The bell rings every 30 minutes starting at 8:00 a.m., so it rings at 8:30 a.m., 9:00 a.m., and so on.

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