Question

The sum of all three digit natural numbers, which are divisible by $7$ is

• A. 60363
• B. 70336
• C. 140672
• D. None of these

Answer 1

Answer: (B)

The smallest and the largest numbers of three digits, which are divisible by $7$ are $105$ and $994$ respectively. So, the sequence of three digit numbers which are divisible by $7$ is $105,112,119,\dots ,994.$ Clearly, it is an $A.P$. with first term $a$ $=105$ and common difference $d=7$. Let there be $n$ terms in this sequence. Then,
$a_n=994$
$\Rightarrow a+(n-1)d=994$
$\Rightarrow 105+(n-1)\times 7=994$
$\Rightarrow n=128$
Now, required sum $=\frac{n}{2}\left[a_1+a_n\right]$
$=\frac{128}{2}[105+994]=70336$