Q1 is basically asking for the list of multiplications which multiply to a certain number.

(i) 4 can be written as 1 x 4 or 2 x 2 (here, 4 x 1 is treated to be the same as 1 x 4)

(ii) 8 can be written as 1 x 8 or 2 x 4.

(iii) 12 can be written as 1 x 12, 2 x 6 or 3 x 4.

(iv) 5 can be written as 1 x 5.

(v) 7 can be written as 1 x 7.

(vi) 1 can be written as 1 x 1.

For Q2, rather than listing out everything, all we need to do is to consider all the possible factors of a given number.

For Q3, the single arrangement indicates that the number of dots is either 1 or a prime number (note that 1 is NOT a prime number).

For Q4, the multiple arrangements indicate that the number of dots is a composite number (basically, numbers which are not 1 and not prime)

But for question 3 and 4 i need rhe explanation
and i dont really understand what you were saying about question 1

For question 1, you need to arrange the dots in a rectangular format.

The number of dots can be thought of as the length * breadth of the rectangle.

We need to find values of “length” and “breadth” such that they multiply to the given number of dots.

For example, 4 dots can be arranged into a length * breadth configuration of 4 * 1 or 2*2.

In the representation,

x x x x

is a configuration of 4 * 1. You can decipher how to draw a 2 * 2 configuration.

I have listed this as the products of the numbers in the previous post.

For question 3, it is a special case of questions 1 and 2 where there is only one possible combination leading to the given number of dots. So for example, 5 dots can be classified as a length * breadth configuration of 1 * 5.

x x x x x

No other combination for 5 dots is possible without leaving gaps in between. The numbers 1 and 5 are technically the only factors of 5 (since you have listed them as the only possible way to arrange the rectangles). As such, these number of dots are essentially prime numbers.

For question 4, it is a special case of questions 1 and 2 where more than one possible arrangement is available.

6 dots, for example, can be arranged as 6 * 1 or 3 * 2. And you have just listed the possible multiplications yielding 6. As such, 6 has more than two factors (1, 2, 3, 6) and these are composite numbers.

So the key idea here is to relate your factors or multiplications listing with the number of dots diagrams.

You will realise that 6 can be written as 1 x 6 and 2 x 3. But why not 4 x 1.5? Because we cannot have 1.5 dots! These are the possible arrangements for the rectangles, since the number of dots along a length or along a breadth must be a whole number.

ok thanks!