1.4 Which number is half way between 17 and 43 ? $$ \qquad $$ $$ \qquad $$ (1) 1.5 Ayesha must repack tins that come in large boxes into smaller boxes. A large box contains 52 tins and a small box can hold 8 tins. Ayesha must repack 90 large boxes. How many small boxes will she need? $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ (2) [7] QUESTION 2 2.1 Express 90 as the product of powers of primes. $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ (2) 2.2 Use prime factors to determine whether $$ \mathbf{9 0} $$ a factor of $$ \mathbf{2 5 0} $$. $$ \qquad $$

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1.4  
To find the number halfway between $$ 17 $$ and $$ 43 $$, we compute their average:  
$$
\frac{17+43}{2}=\frac{60}{2}=30.
$$
So, the number halfway between $$ 17 $$ and $$ 43 $$ is $$ 30 $$.

1.5  
Each large box contains $$ 52 $$ tins. Ayesha has $$ 90 $$ large boxes, so the total number of tins is:  
$$
90 \times 52 = 4680.
$$  
Each small box holds $$ 8 $$ tins. To find the number of small boxes needed, we divide the total tins by $$ 8 $$:  
$$
\frac{4680}{8} = 585.
$$  
Thus, Ayesha will need $$ 585 $$ small boxes.

2.1  
To express $$ 90 $$ as a product of powers of primes, we first find its prime factors:  
$$
90 = 2 \times 45 = 2 \times 3 \times 15 = 2 \times 3 \times 3 \times 5.
$$  
This can be written as:  
$$
90 = 2^1 \times 3^2 \times 5^1.
$$

2.2  
Next, we use prime factors to determine whether $$ 90 $$ is a factor of $$ 250 $$.  
- The prime factorization of $$ 90 $$ is:  
$$
90 = 2^1 \times 3^2 \times 5^1.
$$  
- The prime factorization of $$ 250 $$ is:  
$$
250 = 2 \times 125 = 2 \times 5^3.
$$  
For $$ 90 $$ to be a factor of $$ 250 $$, each prime in $$ 90 $$ must appear in $$ 250 $$ with at least the same exponent.  
- In $$ 90 $$, the exponent of $$ 3 $$ is $$ 2 $$, but $$ 250 $$ does not have $$ 3 $$ as a factor at all.

Since the factor $$ 3^2 $$ is missing in $$ 250 $$, $$ 90 $$ is not a factor of $$ 250 $$.
1.4 The number halfway between $$ 17 $$ and $$ 43 $$ is $$ 30 $$. 1.5 Ayesha will need $$ 585 $$ small boxes. 2.1 $$ 90 = 2^1 \times 3^2 \times 5^1 $$. 2.2 $$ 90 $$ is not a factor of $$ 250 $$.

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