Question

How many multiples of 4 lies between 10 and 205?

Answer 1

Hint: First of all get the least number and the greatest number between the numbers 10 and 205 that are divisible 4. The numbers divisible by 4 are 12, 16, …………., and 204. We can observe that the series 12, 16, …………., and 204, is an arithmetic progression series. Let us assume that the \[{{n}^{th}}\] term of this arithmetic progression is 204. The arithmetic progression has 12 as its first term, 4 as its common difference, and 204 as its \[{{n}^{th}}\] term. Put these values in the formula \[\text{First term}+\left( n-1 \right)\text{common difference}={{n}^{th}}\] term and get the value of n.

Complete step by step solution:
According to the question, we have to find the number of multiples of 4 that are lying between 10 and 205. In other words, we can say that we have to get those numbers between 10 and 205 which are divisible by 4.
The multiples of 4 are 4, 8, 12, 16,………………….
The least number between the numbers 10 and 205 which is divisible by 4 is 12 ………………..………..(1)
To get the next number after 12 which is divisible by 4, we have to add 4 in 12.
Adding 4 in 12, we get
\[4+12=16\]
So, 16 is the next number after 12 which is divisible by 4 ………………………………(2)
We know the property that a number which is divisible by 4 if the last two digits of that number are divisible by 4 ………………………(3)
Using the property shown in equation (3), we can say that 204 is divisible by 4. So, the greatest number between the numbers 10 and 205 which is divisible by 4 is 204 ……………………………(4)
From equation (1), equation (2), and equation (4), we have the numbers divisible by 4.
The numbers which are divisible by 4 are 12, 16, …………………204 ………………………….(5)
We can observe that equation (5) is an Arithmetic progression having a common difference equal to 4. So,
The first term of the Arithmetic progression = 12 ……………………………(6)
The common difference of the Arithmetic progression = \[16-12=4\] ……………………………….(7)
The last term of the Arithmetic progression = 204 ……………………………(8)
Let us assume that the \[{{n}^{th}}\] term of the arithmetic progression is 204 ……………………………(9)
We know the formula, \[\text{First term}+\left( n-1 \right)\text{common difference}={{n}^{th}}\] term……………………………..(10)
Now, putting the value of the first term from equation (6), common difference from equation (7) and \[{{n}^{th}}\] term from equation (9), in the formula shown in equation (10), we get
\[12+\left( n-1 \right)4=204\]
\[\begin{align}
  & \Rightarrow \left( n-1 \right)4=204-12 \\
 & \Rightarrow \left( n-1 \right)4=192 \\
 & \Rightarrow \left( n-1 \right)=\dfrac{192}{4} \\
 & \Rightarrow n-1=48\Rightarrow n=48+1 \\
 & \Rightarrow n=49 \\
\end{align}\]
The \[{{49}^{th}}\] term of the arithmetic progression 12, 16, ………………… is 204.
It means that there are 49 terms in the arithmetic progression 12, 16, ………………… is 204 ……………………..(11)
From equation (11), we have 49 terms in the sequence 12, 16, …………………, and 204.
Therefore, there are 49 terms between the numbers 10 and 205 that are divisible by 4.

Note: In this question, one might think to get all the factors between the numbers 10 and 205, and then count them. But if we do so, then it would be complex as well as time taking to count all the 49 numbers that are divisible by 4.