The-sum-of-two-positive-integers-is-52-and-their-LCM-is-168-Find-the-numbers-

Question Number 19193 by Tinkutara last updated on 06/Aug/17

The sum of two positive integers is 52

and their LCM is 168 . Find the numbers .

Answered by RasheedSindhi last updated on 06/Aug/17

Let the numbers are x and y

and their HCF is h

x + y = 52 \dots \dots . (i)

xy = 168 h \dots \dots (ii)

[∵ product of the numbers = LCM \times HCF]

(i) & (ii) : x + \frac{168 h}{x} = 52

x^{2} - 52 x + 168 h = 0

x = \frac{52 \pm \sqrt{52^{2} - 4 (168 h)}}{2}

= \frac{52 \pm 4 \sqrt{169 - 42 h}}{2}

x = 26 \pm 2 \sqrt{169 - 42 h}

Since x \in Z^{+}, 169 - 42 h is + ve perfect square

and also h > 0 \Rightarrow h = 4

x = 26 \pm 2 \sqrt{169 - 42 \times 4}

= 28, 24

y = 52 - 28, 52 - 24 = 24, 28

Hence the required numbers are

24 and 28

Commented by Tinkutara last updated on 06/Aug/17

Thank you very much Sir!