Number System
Aptitude

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Q.

How many ordered pairs of integer $(x,y)$ are there such that their product is a positive integer less than 100.

 A.

546

 B.

636

 C.

1090

 D.

946

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Solution:
Option(D) is correct

Given $0 < xy < 100$ and $x,y$ are integers.
$x$ and $y$ are either both positive or both negative.

Also given $(x,y)$ is not equal to $(y,x)$.
If $x =1$, $y$ can take values from 1 to 99
⇒ we have $99×2 = 198$ pairs but $(1,1)$ is repeated
Thus can take $198-1 = 197$ pairs.

If $x=2$ , $y$ can take values from 2 to 49
[$(2,1)$ and $(1,2)$ are also covered in 197 pairs above].
⇒ $48×2 - 1 = 95$ pairs [$(2,2)$ is repeated]

Similarly,
If $x=3$ or $y=3$ we have 61 pairs
If $x=4$ or $y=4$ we have 41 pairs
If $x=5$ or $y=5$ we have 29 pairs
If $x=6$ or $y=6$ we have 21 pairs
If $x=7$ or $y=7$ we have 15 pairs
If $x=8$ or $y=8$ we have 9 pairs
If $x=9$ or $y=9$ we have 5 pairs

We have total 473 pairs when $x$ and $y$ are positive.
We will have 473 pairs when $a$ and $b$ are negative.
⇒ We have a total of 946 ordered pairs.


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