Question

The G. C. D. of two numbers is 25% of one of the numbers. The second number is 9 times the G. C. D. If sum of the numbers is 1625, find their L. C.M. ​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

↝ The G. C. D. of two numbers is 25% of one of the numbers.

So,

Let assume that the two numbers be a and b.

↝ G. C. D. be represented by 'g'

So,

$\rm :\longmapsto\:g = 25\% \: of \: a$

$\rm :\longmapsto\:g = \dfrac{25}{100} \times a$

$\rm :\longmapsto\:g = \dfrac{1}{4} \times a$

$\rm \implies\:\boxed{ \tt{ \: a \: = \: 4g \: }} - - - (1)$

Also, given that,

↝ The second number is 9 times the G. C. D.

So,

$\rm \implies\:\boxed{ \tt{ \: b \: = \: 9g \: }} - - - (2)$

Further, given that,

↝ Sum of the numbers is 1625.

$\rm :\longmapsto\:4g + 9g = 1625$

$\rm :\longmapsto\:13g = 1625$

$\rm :\longmapsto\:g = \dfrac{1625}{13}$

$\bf\implies \:\boxed{ \tt{ \: \: g \: = \: 125 \: \: }}$

Now,

We know that,

L. C. M (a, b) × G. C. D (a, b) = a × b

So, on substituting the values, we get

$\rm :\longmapsto\:L. C. M \times g = a \times b$

$\rm :\longmapsto\:L. C. M \times g = 4g \times 9g$

$\rm :\longmapsto\:L. C. M = 36g$

$\rm :\longmapsto\:L. C. M = 36 \times 125$

$\rm \implies\:\boxed{ \tt{ \: L. C. M \: = \: 4500 \: \: }}$