Question

: How many three-digit numbers are divisible by 4?

Answer 1

Hint: We already know that 100 is the smallest three-digit number and 999 is the highest. So, the first step should be to determine the integers that are the closest to these and are divisible by 4. We'll use the arithmetic progression after determining the smallest and largest three-digit numbers divisible by four.

Complete step by step answer:
To determine the smallest 3-digit number divisible by 4, we will divide 100 with the number 4.
\[ \Rightarrow \dfrac{{100}}{4} = 25\], that means 100 is purely divisible by 4.
After this, to determine the biggest 3-digit number divisible by 4, we will divide 999 with 4,
\[ \Rightarrow \dfrac{{999}}{4} = 249\dfrac{3}{4}\], as we can see 3 is the remainder, therefore we will subtract 3 from 999 to get the number divisible by 4.
\[ \Rightarrow 999 - 3 = 996\]
Hence 996 is the biggest 3-digit number divisible by 4.
Since it is an Arithmetic progression. We can simply determine the numbers by subtracting the first term from the last term.
We know that the smallest number divisible by 4 is 100 which implies that the first term is the 25th term. However, we will deduct 1 from 25 because we must also consider 100.
Therefore, \[\left( {25 - 1} \right)th{\text{ term}} = 24th{\text{ term}}\]
Now, our last term will be the 249th term which comes out when 996 is divided by 4.
Hence, the total numbers lying between 100 and 996 is
\[ \Rightarrow (249 - 24) = 225\] terms
Therefore, there are 225 terms lying between the smallest and the biggest 3- digit natural numbers divisible by 4.

Note:
 The other way by using Arithmetic Progression series is by using the arithmetic progression formula \[l = a + (n - 1)d\] where the first term (a) is 100, the last term (l) is 996, the common difference (d) is 4 and n is the number lying between them. Therefore,
\[l = a + (n - 1)d\]
Now by replacing \[l = 996\], \[a = 100\], and \[d = 4\]we get
\[ \Rightarrow 996 = 100 + \left( {n - 1} \right)4\]
\[ \Rightarrow 896 = \left( {n - 1} \right)4\]
\[ \Rightarrow 224 = n - 1\]
Therefore,
\[ \Rightarrow n = 225\]
Hence, by using both the methods there are 225 three-digit natural numbers completely divisible by 4.