2.1. Write down the HCF of 72 and 120 . 2.2. Write 1485 as a product of its prime factors. Show your working. 2.3. Determine the sum of all the factors of 100 2.4. Show that 899 is not a prime number. 2.5. Divide 240 in the ratio $$ 5: 3: 4 $$ 2.6. Ntsako's car uses 1 litre of fuel to travel 12 km . How much fuel will be needed to 2.7. A certain distance is covered in 3 hours at an average speed of $$ 120 \mathrm{~km} / \mathrm{h} $$. How Long will it take to cover the same distance at an average speed of $$ 90 \mathrm{~km} / \mathrm{h} $$ ?

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2.1.  
The prime factorization of $$72$$ is  
$$
72=2^3\times3^2,
$$  
and that of $$120$$ is  
$$
120=2^3\times3\times5.
$$  
The HCF is found by taking the lowest powers of the common prime factors:  
$$
\text{HCF}=2^3\times3=8\times3=24.
$$

2.2.  
Factorize $$1485$$ as follows:  
$$
1485\div3=495,
$$  
$$
495\div3=165,
$$  
$$
165\div3=55,
$$  
and  
$$
55=5\times11.
$$  
Thus, the prime factorization is  
$$
1485=3\times3\times3\times5\times11=3^3\times5\times11.
$$

2.3.  
First, write $$100$$ as a product of primes:  
$$
100=2^2\times5^2.
$$  
The sum of all factors of a number $$n=p^a\times q^b$$ is given by  
$$
\sigma(n)=\left(\frac{p^{a+1}-1}{p-1}\right)\left(\frac{q^{b+1}-1}{q-1}\right).
$$  
For $$100$$:  
$$
\sigma(100)=\left(\frac{2^{2+1}-1}{2-1}\right)\left(\frac{5^{2+1}-1}{5-1}\right)
=\left(\frac{2^3-1}{1}\right)\left(\frac{5^3-1}{4}\right)
=\left(8-1\right)\left(\frac{125-1}{4}\right).
$$  
This simplifies to:  
$$
7\times\frac{124}{4}=7\times31=217.
$$

2.4.  
To show that $$899$$ is not a prime number, we look for factors other than $$1$$ and $$899$$. Notice that  
$$
29\times31=899,
$$  
so $$899$$ can be factored as  
$$
899=29\times31.
$$  
Thus, $$899$$ is not a prime number.

2.5.  
We need to divide $$240$$ in the ratio $$5:3:4$$. First, find the total number of parts:  
$$
5+3+4=12.
$$  
Then, compute each share:  
$$
\text{First share}=\frac{5}{12}\times240=100,
$$  
$$
\text{Second share}=\frac{3}{12}\times240=60,
$$  
$$
\text{Third share}=\frac{4}{12}\times240=80.
$$

2.6.  
Ntsako's car uses $$1$$ litre of fuel to travel $$12$$ km. Therefore, to travel a distance $$d$$ km, the fuel needed is given by  
$$
\text{Fuel}=\frac{d}{12}\text{ litres}.
$$  
(If a specific distance were given, you would substitute it for $$d$$. For example, to travel $$60$$ km, the required fuel is  
$$
\frac{60}{12}=5\text{ litres}.
$$)

2.7.  
First, determine the distance covered in $$3$$ hours at $$120$$ km/h:  
$$
\text{Distance}=120\times3=360\text{ km}.
$$  
Now, to cover $$360$$ km at $$90$$ km/h, the time required is  
$$
\text{Time}=\frac{360}{90}=4\text{ hours}.
$$
2.1. The HCF of 72 and 120 is 24. 2.2. 1485 can be factored into prime numbers as $$3^3 \times 5 \times 11$$. 2.3. The sum of all factors of 100 is 217. 2.4. 899 is not a prime number because it can be divided by 29 and 31. 2.5. Dividing 240 in the ratio 5:3:4 gives 100, 60, and 80 respectively. 2.6. Ntsako's car uses $$\frac{d}{12}$$ litres of fuel to travel $$d$$ km. 2.7. It will take 4 hours to cover the same distance at 90 km/h.

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