How many three-digit natural numbers are divisible by 7? - Vidyadip

Question

How many three-digit natural numbers are divisible by $7$?

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Answer

The three-digit natural numbers divisible by $7$ are $105, 112, 119, ..., 994$.
Clearly, three number are in AP.
Here, $a=105$ and $d=112-105=7$
Let this AP contains $n$ terms. Then,
$a_n=994$
$\Rightarrow 105+(n-1) \times 7=994$
$\Rightarrow 7 n+98=994\left[a_n=a+(n-1) d\right]$
$\Rightarrow 7 n=994-98=896$
$\Rightarrow n=128$
Hence, there are $128$ three-digit numbers divisible by $7$.

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