The L.C.M. and ratio of four numbers are 630 and 2 : 3 : 5 : 7 respectively. The difference between the greatest and least numbers is

### Concept & Logic The L.C.M. of numbers given in a ratio where the ratio terms are mutually coprime is simply the product of those ratio parts multiplied by their Highest Common Factor (H.C.F.). $$ \text{L.C.M.} = \text{H.C.F.} \times (\text{Product of coprime ratio parts}) $$ ### Step-by-Step Solution * Let the common factor (H.C.F.) be $x$. The four numbers can be represented as $2x$, $3x$, $5x$, and $7x$. * Because the numbers 2, 3, 5, and 7 are prime to each other (coprime), their L.C.M. is the product of these ratios and $x$. * L.C.M. = $2 \times 3 \times 5 \times 7 \times x = 210x$. * We are given that the L.C.M. is 630. * Equating the two: $210x = 630$. * Solving for $x$: $x = 630 / 210 = 3$. * The greatest number is $7x = 7 \times 3 = 21$. * The least number is $2x = 2 \times 3 = 6$. * The difference between the greatest and least numbers is $21 - 6 = 15$. ### Exam Strategy & Shortcut Look at the ratio: the difference between the greatest part (7) and the least part (2) is 5 parts. This means the final answer MUST be a multiple of 5. Scanning the given options (6, 14, 15, 21), 15 is the only multiple of 5. You can confidently select it without calculating the L.C.M. or solving for $x$. ### Common Pitfall A frequent mistake is solving for $x = 3$ and then mistakenly choosing option (a) which is 6, or option (d) which is 21, by forgetting that the question specifically asks for the *difference* between the two extreme numbers, not just one of the numbers. ### Final Answer **Therefore, the correct answer is 15.**