Two numbers are in ratio 2:7 and their L.C.M. is 182. The greater number is

Two numbers are in ratio 2:7, so they can be written as:

First number $= 2x$

Second number $= 7x$

where $x$ is a common multiplier.

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Since 2 and 7 are coprime (no common factors), the L.C.M. of $2x$ and $7x$ is:

L.C.M. $= 2 \times 7 \times x$

L.C.M. $= 14x$

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Given that L.C.M. $= 182$:

$14x = 182$

$x = \dfrac{182}{14}$

$x = 13$

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The two numbers are:

First number $= 2x = 2 \times 13 = 26$

Second number $= 7x = 7 \times 13 = 91$

The greater number is $91$.

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Therefore, the answer is option 2: $91$