Question

The LCM of the two natural numbers is 24. If their sum is 14, find the numbers.
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Answer 1

The given two numbers are 2 and 12.

• The L.C.M. of two numbers is given as 24.

• Let the two numbers be x and y.

• The sum of the numbers is x + y.

According to given,

x + y = 14 -(i)

• Now, L.C.M. stands for the lowest common multiple of any two or more numbers.

• Here, the L.C.M. is 24. It means that that the product of x and y will be 24 (refer to the image attached below).

=> xy = 24

=> x = 24 / y -(ii)

• Putting x as 24 / y in equation (i), we get,

(24 / y) + y = 14

=> (24 + y²) / 24 = 14

=> 24 + y² = 14 y

=> y² - 14y - 24 = 0

=> y² - 12y - 2y - 24 = 0

=> y ( y - 12 ) - 2 (y - 12) = 0

=> ( y - 2) ( y - 12) = 0

=> y - 2 = 0, y - 12 = 0

=> y = 2, y = 12

• Putting the value in equation (ii), we get,

If y = 2, x = 24 / 2 = 12

If y = 12, x = 24 / 12 = 2

• Therefore, the given two numbers are 2 and 12.

Answer 2

The two numbers are 8 and 6

Explanation:

Let x and y represents the two natural numbers.

Thus, the two equation is given by

$xy=24d$ (By LCM rule)

$x+y=14$

     $y=14-x$

Substituting $y=14-x$ in the equation $xy=24d$, we get,

$x(14-x)=24d$

Multiplying the terms, we have,

           $14x-x^2=24d$

$-x^2+14x-24d=0$

  $x^2-14x+24d=0$

Let us solve the equation using the quadratic formula.

Thus, we have,

$x=\frac{14\pm \sqrt{14^2-4(1)(24d)}}{2(1)}$

$x=\frac{14\pm \sqrt{196-96d}}{2}$

$x=\frac{14\pm \sqrt{4(49-24d)}}{2}$

$x=\frac{14\pm 2 \sqrt{49-24d}}{2}$

We need to determine the value of d, such that the expression $\swrt{49-24d}$ is a perfect square.

Let us substitute d = 2, we get,

$x=\frac{14\pm 2 \sqrt{49-24(2)}}{2}$

$x=\frac{14\pm 2 \sqrt{49-48}}{2}$

$x=\frac{14\pm 2 \sqrt{1}}{2}$

$x=\frac{14\pm 2 }{2}$

Hence, the values of x are

$x=\frac{14+ 2}{2}$ and $x=\frac{14-2}{2}$

$x=\frac{16}{2}$  and  $x=\frac{12}{2}$

$x=8$  and  $x=6$

Therefore, the values of x are 8 and 6

Hence, the two numbers are 8 and 6

Learn more:

(1) The LCM of the two number is 24 their sum 14 what are the numers ​

brainly.in/question/9075433

(2) The lcm of two natural number is 24. If their sum is 14,find the number.​

brainly.in/question/15365435