The least common multiple and the highest common factor of two positiv

Question

The least common multiple and the highest common factor of two positive integers x and y are 105 and 3, respectively. If x + y = 36, the what is the value of 1/x + 1/y ?

(A) 1 
(B) 4/35
(C) 1/6
(D) 12/315 
(E) 1/315

iridescent995 · Answer

x*y = 3*105
x+y = 36

1/(1/x + 1/y) = 3*105/36

1/x+1/y = 4/35

shivxlon · Answer

HCF(X,Y) = 3
LCM(X,Y) = 105
also, X+Y = 36

since, we know that X*Y = LCM*HCF, therefore X*Y=3*105

thus 1/x+1/y = (x+y)/x*y = 36/(3*105) = 4/35  (B)

sj296 · Answer

Hey @Bunnel, the lcm of two positive integers is always greater than HCF, right? so, in this question HCF should be 3 and LCM should be 105.

However, it wouldn't change the answer.

IanStewart · Answer

That will be true almost always, but technically, that's not quite right -- the HCF and LCM can also be equal, when your numbers are all the same. The LCM of 25 and 25 is equal to 25, and the HCF of 25 and 25 is also equal to 25, for example. You're very right that there's an error in the question though!

Bunuel · Answer

Yes, you are right. Edited. Thank you.

MathRevolution · Answer

Product of two numbers (x and y) = LCM (x,y) * HCF (x,y)

=> xy = 105 * 3

=>  \frac{1}{x} + \frac{1}{y} = \frac{x + y }{ xy} = \frac{36 }{ (105 * 3)} = \frac{12}{105} = \frac{4}{35}

Answer B

BrushMyQuant · Answer

Theory

➡ Product of two numbers = Product of their LCM and HCF

The least common multiple and the highest common factor of two positive integers x and y are 105 and 3

LCM = 105, HCF = 3
Product of the numbers = x*y = LCM * HCF = 105 * 3 = 315 ....(1)

If x + y = 36, the what is the value of 1/x + 1/y

x + y = 36
Divide both the sides by xy we get

\frac{x + y}{xy}  =  \frac{36}{xy} 
=>  \frac{x}{xy}  +  \frac{y}{xy}  =  \frac{36}{315}  (as xy = 315 from (1) )
=>  \frac{1}{y}  +  \frac{1}{x}  =  \frac{4}{35}

So,  Answer will be B .
Hope it helps!

Watch the following video to Learn the Basics of LCM and GCD

https://www.youtube.com/watch?v=7jPV6JD0908

brivibas39tv · Answer

Alternative method:

Prime factorize 105 -> 3 x 5 x 7
Clearly, therefore, we see that for x + y to be 36, x = 21 (3 x 7) and y = 15 (3 x 5)
GCF will also be 3.
After that just do the addition.

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