I have two questions from the chapter, Number Properties.

**1)**

Question:

Answer:

Although I got the answer for this one by listing multiples and picking the ones that are multiples of both 2 and 7 to get 8 numbers as the answer, I see that they have used another method that involves a “17th multiples of 14” and subtracting 10 from 17 and adding 1 for inclusiveness. I wish someone could explain what is happening in this part.

**2)**

Question:

Answer:

In this question’s answer, how do we calculate the range? I am aware that

Range = Biggest number - Smallest number

I don’t understand why the range here is 2 if they are consecutive. It must be 1?

Considering the example given, if a = -10 and the next consecutive number towards right is b which is -9, the range will be:

Range = -9 - (-10) = - 9 + 10 = 1

How did 2 come here? I might need an explanation for this too

Thanks in advance!

- For this question,

observe that, we need to find multiples of 2 and 7 from 140 to 240, i.e,

in reality, we need to find the number of multiples of (7x2) = 14, present within 140 to 240(inclusive).

Now, starting number 140 is a multiple of 14, but the ending number 240 is not.

So, to find out easily what is the last number which is a multiple of 14 easily, we divide the last given number by 14, i.e, in our case, (240/14)= 17.14. Here, as we receive the remainder, we ignore that and just take 14 and 17 and multiply them, i.e, (14x17)=238, to receive the complete number which is totally a multiple of 14, without any kind of remainder.

Therefore, the actual range of numbers becomes from 140 to 238, where 1st number and last number in the series are multiple of both 2 and 7 or 14.

The formula for the Total Number of Numbers within a series is:

(Last Number - 1st Number) + 1;

But if we directly apply this here, we get,

(238 - 140) + 1 = 99 numbers, i.e, [140,141,142,…,235,…,238]

But, we don’t need this we need multiples of 14, from 140 to 238

Now, to obtain this:

[(238 - 140)/14] +1 = 8 numbers total.

- among, a =-10, b=-9 and c=-8,

**c** is the largest, and **a** is the smallest, that is why,

Range = -8-(-10) = 2, range will always be 2.