Question

Find the number of integers between $ 10 $ and $ 40 $ , inclusive, which leave a remainder of zero when divided by $ 3 $ .

Answer 1

Hint: According to the question we have to find such numbers which leaves the remainder of zero when divided by $ 3 $ . It means that the numbers are completely divisible by $ 3 $ , so we can say that it is a multiple of $ 3 $ . WE will find out all the multiples between $ 10 $ and $ 40 $ , and that gives us the answer.

Complete step by step solution:
We have to find the number of integers between
 $ 10 $ and $ 40 $ .
We know that integers are the set of whole numbers, the set of natural numbers also called counting numbers. These numbers can either be positive, negative or zero but it cannot be a fraction. As for example:
 $ 1,2,3, - 1, - 2, - 3... $
These are all positive and negative integers.
Now according to the question it is given that we have to find the integer which leaves a remainder of zero when divided by $ 3 $ .
So a number when divided by $ 3 $ leaves a remainder $ 0 $ , it is a multiple of $ 3 $ .
Now we will find all the multiples between
  $ 10 $ and $ 40 $ .
Therefore the multiples of $ 3 $ between $ 10 $ and $ 40 $ are:
 $ 12,15,18,21,24,27,30,33,36,39 $
These all are the divisible of $ 3 $ that lies between $ 10 $ and $ 40 $ .
The total numbers of these are
 $ = 10 $ .
Hence there are total $ 10 $ integers,
So, the correct answer is “ $ 10 $ integers”.

Note: We should also know the divisibility test of $ 3 $ . The divisibility rule of $ 3 $ says that – A number is divisible by $ 3 $ , if the sum of all its digits is a multiple of $ 3 $ or divisibility of $ 3 $ . Let us take an example-
 (a) $ 54 $
We will sum all of its digits i.e.
 $ 5 + 4 = 9 $
Now we can say that $ 9 $ is divisible by $ 3 $ , so
 $ 54 $ is divisible by $ 3 $ .