Question

The number of ordered pairs $(x,y)$ such that $LCM$ of $x$ and $y$ is $2120$. $x,y$ are positive integers

• A. $105$
• B. $150$
• C. $315$
• D. $210$

Answer 1

The correct option is C $315$

Given that,

$2120=2\times 2\times 2\times 3\times 3\times 5\times 7$

$2120=2^3\times 3^2\times 5\times 7$

the various ways to distribute $2^3$ to

(x,y) are

$(2^3,1),(2^3,2),(2^3,2^2),(2^3,2^3),(1,2^3),$

$(2,2^3),(2^2,2^3)$

Which count to be $2\times 3+1=7$

Similarly, others factors can be distributed

to x and y.

thus the total numbers of ways is

$(2\times 3+1)(2\times 2+1)(2\times 1+1)(2\times 1+1)=315$

Hence, this is the required answer.