Question

If $ n $ is a factor of 72, such that $ xy = n $ , then the number of ordered pairs $ (x,y) $ are: (Where $ x,y \in \mathbb{N} $ )
(A) 40
(B) 50
(C) 60
(D) 70

Answer 1

Hint: Factor is the number that divides the given number. Therefore, $ n $ will be the collection of all the possible numbers that divide 72. Find all those numbers. Then equate each number with $ xy $ and find the possible values of $ (x,y) $ such that $ xy = n $

Complete step-by-step answer:
It is given in the question that, $ n $ is a factor of 72.
Thus, possible values of $ n $ can be written as
 $ \Rightarrow n = \{ 1,2,3,4,6,8,9,12,18,24,36,72\} $
Now, it is also given that
 $ xy = n $
We need to find the ordered pairs $ (x,y) $ such that their product is equal to $ n $ . For that, we will consider one value of $ n $ and find the possible values of $ (x,y) $ such that $ xy = n $ .
Then we will take another value of $ n $ and find the possible values of $ (x,y) $ such that $ xy = n $ .
We will keep doing this until we exhaust all the values of $ n $ .
For $ n = 1 $
 $ (x,y) = \{ (1,1)\} $
For $ n = 2 $
 $ (x,y) = \{ (1,2),(2,1)\} $
For $ n = 3 $
 $ (x,y) = \{ (1,3),(3,1)\} $
For $ n = 4 $
 $ (x,y) = \{ (1,4),(4,1),(2,2)\} $
For $ n = 6 $
 $ (x,y) = \{ (1,6),(6,1),(2,3),(3,2)\} $
For $ n = 8 $
 $ (x,y) = \{ (1,8),(8,1),(2,4),(4,2)\} $
For $ n = 9 $
 $ (x,y) = \{ (1,9),(9,1),(3,3)\} $
For $ n = 12 $
 $ (x,y) = \{ (1,12),(12,1),(3,4),(4,3),(2,6),(6,2)\} $
For $ n = 18 $
 $ (x,y) = \{ (1,18),(18,1),(2,9),(3,6),(9,2),(6,3)\} $
For $ n = 24 $
 $ (x,y) = \{ (1,24),(24,1),(2,12),(12,2),(3,8),(8,3),(4,6),(6,4)\} $
For $ n = 36 $
 $ (x,y) = \{ (1,36),(36,1),(2,18),(18,2),(3,12),(12,3),(4,9),(9,4),(6,6)\} $
For $ n = 72 $
 $ (x,y) = \{ (1,72),(72,1),(2,36),(36,2),(3,24),(24,3),(4,18),(18,4),(6,12),(12,6),(8,9),(9,8)\} $
Now, when we calculate all the possible values of $ (x,y) $ , we get
 $ \Rightarrow $ The number of ordered pairs $ (x,y) = 60 $
Therefore, from the above explanation, the correct answer is, option (C) 60
So, the correct answer is “Option C”.

Note: This question gets very lengthy. In such cases you need to be careful to not make any calculation mistakes or to skip any number. We can reduce the solution by writing the ordered pairs in one single order and then multiplying it by 2. You just need to be careful to exclude the ordered pairs of the form $ (a,a) $ while multiplying by 2 as they will not have two different ordered pairs.