Question
How many numbers are there between $101$ and $999$, which are divisible by both $2$ and $5$?
✓
Answer
For the number to be divisible by both 2 and 5, they have to be divisible by the LCM of $2$ and $5 = 10$.
The numbers divisible by 10 between 101 and 999
Are $110, 120, 130, ...., 990$
Here
$a = 110$
$d = 10$
$a_n = a + (n - 1)d$
$\Rightarrow 990 = 110 + (n - 1)(10)$
$\Rightarrow 990 = 110 + 10n - 10$
$\Rightarrow 890 = 10n$
$\Rightarrow n = 89$
Thus, $89$ numbers between $101$ and $999$ are divisible by both $2$ and $5$.
The numbers divisible by 10 between 101 and 999
Are $110, 120, 130, ...., 990$
Here
$a = 110$
$d = 10$
$a_n = a + (n - 1)d$
$\Rightarrow 990 = 110 + (n - 1)(10)$
$\Rightarrow 990 = 110 + 10n - 10$
$\Rightarrow 890 = 10n$
$\Rightarrow n = 89$
Thus, $89$ numbers between $101$ and $999$ are divisible by both $2$ and $5$.
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