Gujarat BoardEnglish MediumSTD 10MathsArithmetic Progression4 Marks
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,
$ \mathrm{a}_{\mathrm{n}}=994 $
$ \Rightarrow 105+(\mathrm{n}-1) \times 7=994 $
$ \Rightarrow 7 \mathrm{n}+98=994\left[\mathrm{a}_{\mathrm{n}}=\mathrm{a}+(\mathrm{n}-1) \mathrm{d}\right] $
$ \Rightarrow 7 \mathrm{n}=994-98=896 $
$ \Rightarrow \mathrm{n}=128$
Hence, there are $128$ three-digit numbers divisible by $7.$
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