Question

Find the number of numbers from 1to 200 divisible by 3 and 4

Answer 1

$\boxed{\boxed{\large\mathbf{16}}}$
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$\underline{\underline{\huge\mathfrak{ANSWER}}}$

We need to find LCM of 3, 4

Any no. divisible by LCM of 3, 4 will be divisible by 3, 4

$\mathbf{LCM \ = 12}$

Now,

Least number divisible by 12 between 1 and 200 is 12.

Highest number between 1 and 200 divisible by 12 is

$\textsf{Dividing 200 by 12 and subtracting the remainder from 200}$

On dividing 200 by 12, we get remainder as $\bf{8}$

Highest number divisible by 12 between 1 and 200

$\mathsf{\Longrightarrow 200 - 8 = 192}$

So,

In A.P 12, 24, 36........192

$\mathsf{a = 12}$

$\mathsf{l(last \: term) = 192}$

$\mathsf{d(common \: difference) = 12}$

$\mathsf{n(no. \: of \: terms) = ?}$

$\underline\text{Using the Formula}$

$\boxed{\mathbf{a_n = a + (n-1)d}}$

$\mathsf{\Longrightarrow 12 + (n - 1)12 = 192}$

$\mathsf{\Longrightarrow 12(n - 1) = 192 - 12}$

$\mathsf{\Longrightarrow (n - 1) = \large{\frac{\cancel{180}}{\cancel{12}}}}$

$\mathsf{\Longrightarrow n - 1 = 15}$

$\Longrightarrow \mathbf{n = 16}$

Hence, there are 16 numbers which are divisible by both 3 and 4 between 1 and 200.
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