12 (a) $$ 50=2 \times 5^{2} $$ $$ 125000=50^{3} $$ $$ 125000=2^{a} \times 5^{b} $$ Find $$ a $$ and $$ b $$. Answer $$ a= $$ $$ \qquad $$ $$ b= $$ $$ \qquad $$ [1] (b) $$ 9800=2^{3} \times 5^{2} \times 7^{2} \quad 7875=3^{2} \times 5^{3} \times 7 $$ Write as a product of its prime factors (i) the highest common factor (HCF) of 9800 and 7875, Answer $$ \qquad $$ [1] (ii) the lowest common multiple (LCM) of 9800 and 7875 .

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Let's solve part (a) first.

We have the equations:
1. $$ 50 = 2 \times 5^{2} $$
2. $$ 125000 = 50^{3} $$
3. $$ 125000 = 2^{a} \times 5^{b} $$

### Step 1: Calculate $$ 50^{3} $$

Using the first equation, we can express $$ 50 $$ in terms of its prime factors:
$$
50 = 2^{1} \times 5^{2}
$$
Now, we can calculate $$ 50^{3} $$:
$$
50^{3} = (2^{1} \times 5^{2})^{3} = 2^{3} \times 5^{6}
$$

### Step 2: Set $$ 125000 $$ equal to $$ 2^{a} \times 5^{b} $$

From the second equation, we have:
$$
125000 = 50^{3} = 2^{3} \times 5^{6}
$$
Thus, we can equate this to the third equation:
$$
125000 = 2^{a} \times 5^{b}
$$
This gives us:
$$
a = 3 \quad \text{and} \quad b = 6
$$

So, the answers for part (a) are:
- $$ a = 3 $$
- $$ b = 6 $$

Now, let's move on to part (b).

### Part (b)

We need to find:
(i) the highest common factor (HCF) of $$ 9800 $$ and $$ 7875 $$,
(ii) the lowest common multiple (LCM) of $$ 9800 $$ and $$ 7875 $$.

#### Step 1: Prime Factorization

We have:
- $$ 9800 = 2^{3} \times 5^{2} \times 7^{2} $$
- $$ 7875 = 3^{2} \times 5^{3} \times 7^{1} $$

#### Step 2: Find HCF

To find the HCF, we take the lowest power of each common prime factor:
- For $$ 2 $$: $$ \min(3, 0) = 0 $$ (not present in 7875)
- For $$ 3 $$: $$ \min(0, 2) = 0 $$ (not present in 9800)
- For $$ 5 $$: $$ \min(2, 3) = 2 $$
- For $$ 7 $$: $$ \min(2, 1) = 1 $$

Thus, the HCF is:
$$
HCF = 5^{2} \times 7^{1} = 25 \times 7 = 175
$$

#### Step 3: Find LCM

To find the LCM, we take the highest power of each prime factor:
- For $$ 2 $$: $$ \max(3, 0) = 3 $$
- For $$ 3 $$: $$ \max(0, 2) = 2 $$
- For $$ 5 $$: $$ \max(2, 3) = 3 $$
- For $$ 7 $$: $$ \max(2, 1) = 2 $$

Thus, the LCM is:
$$
LCM = 2^{3} \times 3^{2} \times 5^{3} \times 7^{2}
$$

Now, let's calculate the LCM:
$$
LCM = 8 \times 9 \times 125 \times 49
$$

Calculating this step by step:
1. $$ 8 \times 9 = 72 $$
2. $$ 72 \times 125 = 9000 $$
3. $$ 9000 \times 49 = 441000 $$

So, the answers for part (b) are:
- (i) HCF = 175
- (ii) LCM = 441000

### Final Answers:
(a) $$ a = 3 $$ \quad $$ b = 6 $$ \
(b) (i) HCF = 175 \
(ii) LCM = 441000
$$ a = 3 $$, $$ b = 6 $$, HCF = 175, LCM = 441000

12 (a)

Find and .

Answer [1]
(b)

Write as a product of its prime factors
(i) the highest common factor (HCF) of 9800 and 7875,

Answer
[1]
(ii) the lowest common multiple (LCM) of 9800 and 7875 .

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12 (a) Find and . Answer [1] (b) Write as a product of its prime factors (i) the highest common factor (HCF) of 9800 and 7875, Answer [1] (ii) the lowest common multiple (LCM) of 9800 and 7875 .