Question

The Least Common multiple (L.C.M) of two numbers is 144 and their Greatest Common
Divisor (G.C.D) is 2. What are the numbers? Given their sum is 34.

Answer 1

Let a no. be x

then,

the other no. be 34 - x

as, sum of the the no. is 34..

now,

$a \times b = lcm(a \: and \: b) \times hcf(a \: and \: b)$

using this formula,

$x(34 - x) = 144 \times 2$

$34x - {x}^{2} = 288$

${x}^{2} - 34x + 288 = 0$

${x}^{2} - 18x - 16x + 288 = 0$

$x( x - 18) - 16(x - 18) = 0$

$(x - 18)(x - 16) = 0$

$x = 18 \: or \: 16$

when x = 18

$first \: no \: = 18 \\ second \: no = 16$

when x = 16

$first \: no \: = 16 \\ second \: no. = 18$

so,

the two no. are 18 and 16