Question

Find the number of natural numbers between 102 and 998 which are divisible by 2 and 5 both.

Answer 1

Hint: We are required to find the number of natural numbers between two given natural numbers with a condition that those numbers should be divisible by 2 and 5 both. We should be aware about the concept of divisibility. Moreover, we would use the fact that if a number is divisible by both 2 and 5 then it can be said the number is divisible by 10.

Complete step-by-step solution:
We have a range of numbers varying from 102 to 998. We need to find a number divisible by 2 and 5 both. Suppose you have a number$x$, then if it divisible by 2 and 5 both then we can write:
$x=2\times r\times 5\times t$
For some natural numbers $r$ and $t$
$\implies x=10\times k$
Where$k=r\times t$, which is another natural number.
So, we can say that the number should be divisible by 10, which will be the following numbers:
 110,120,$\ldots$, 990.
We now use the arithmetic progression formulae:
The first term,$a=110$
$n^{th}$ term,$a_n=990$
And the common difference,$d=10$
Now, we use the formula:
$a_n=a+\left(n-1\right)\times d$
Putting the values:
$990=110+\left (n-1\right)\times 10$
$\implies 880=10n-10$
$\implies 890=10n$
$\implies n=89$
Hence, there are 89 natural numbers between 102 and 998 which are divisible bye 2 and 5 both.

Note: Do not write all natural numbers between 102 and 998 and then circle the ones divisible by 2 and then by 5 and then find the intersection of both, because that would take up a lot of time. Using the simple technique of arithmetic progression would be a better option. Also, note that when we are asked to find numbers ‘between’ two numbers, then the end numbers are not included.