   Question Answers

# The LCM of two numbers is 567 and their HCF is 9. If the difference between the two numbers is 18, find the two numbers.(A) 36 and 18(B) 78 and 60(C) 63 and 81(D) 52 and 34  Hint: To solve this particular type of question, use the fact that the product of two numbers is equal to the product of LCM and HCF of those two numbers. Express the numbers in the form of a variable x and then use the the above property to find the desired result.

Complete step-by-step solution:
Product of two number = LCM$\times$HCF= 567×9 = 5103
Let the two numbers be x ( smaller number ) and x+18 ( larger number)
x( x+18 ) = 5103
$\Rightarrow {x^2} + 19x - 5103 = 0 \\ \Rightarrow {x^2} + 81x - 63x - 5103 = 0 \\ \Rightarrow x\left( {x + 81} \right) - 63(x + 81) = 0 \\ \Rightarrow \left( {x - 63} \right)\left( {x + 81} \right) = 0 \\$
Therefore x = 63
(neglecting the negative quantity )
And another number = x+18 =81.

Note: Remember to recall the basic properties of LCM and HCF to use in these kinds of questions. Also remember to use the "splitting method" to solve the quadratic equation we got in the last. Use factorization of 5103 as it is a large quantity to find the middle terms. Note that if we had taken both numbers as x and x-18 , then too we would have got the same answers.
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