Question

the sum and least common multiple of two positive integers x,y are given as x+y=40 and LCM [x,y] = 48. find the numbers x and y.

Answer 1

SOLUTION

GIVEN

The sum and least common multiple of two positive integers x,y are given as x + y = 40 and LCM [x,y] = 48

TO DETERMINE

The number x and y

EVALUATION

Here it is given that the sum and least common multiple of two positive integers x,y are given as x + y = 40 and LCM [x,y] = 48

Let HCF [x,y] = k

Then

LCM × HCF = Product of the numbers

∴ Product of the numbers

= xy

= LCM × HCF

= 48k

Again x + y = 40

So x and y are the roots of the quadratic equation

$\sf{ {t}^{2} - 40t + 48k = 0 }$

Thus we see that sum of the zeroes is 40 and product of the zeroes is multiple of 48

Which holds when the roots are 24 and 16.

Consequently k = 8

Hence x = 24 , y = 16 or x = 16 , y = 24

FINAL ANSWER

x = 24 , y = 16 or x = 16 , y = 24

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