In other words, how many distinct values can x/y have if x

In other words, how many distinct values can x/y have if x and y are both integers from 1 to 10?

(A) 63 (B) 65 (C) 67 (D) 69 (E) 71

My apologies for the error in the original!

total = 100

1/1 or 1 = 10 times but count only once.
1/2 = 2/4 = 4/8 = 3/6 = 5/10 = 5 times so count only once.
1/3 = 2/6 = 3/9 = 3 times but count only once.
.
.
.
.
.
10/9 = only once

i willl correct if any, i found 67.

eager to know the elegant approach......

I tried this. But my method was very time consuming. What is the source of this problem?

All possible combinations - 10*10 = 100.
Now start pairing possible duplicates:
(1/2,2/4,3/6,4/8,5/10) and their reciprocals.
(1/3,2/6,3/9) and their reciprocals.
(1/4,2/8) and their reciprocals.
(1/5,2/10) and their reciprocals.
(2/3,4/6,6/9) and their reciprocals.
(2/5,4/10) and their reciprocals.
(3/4,6/8) and their reciprocals.
(3/5,6/10) and their reciprocals.
(1/1,2/2,....10/10).

Subtract and get the answer. I am feeling too tired to do so!

Hint: You would need to consider at least one member from each group and the corresponding group of its reciprocals.

I posted a solution.

https://www.gmatclub.com/phpbb/viewtopic.php?t=31482

Is that correct?

you are missing 4/5, 8/10 and their reciprocals.

but i guess it is 65.
1/2 5 4
1/3 3 2
1/4 2 1
1/5 2 1
2/3 2 1
2/5 2 1
3/4 2 1
3/5 2 1
4/5 2 1
13
the reciprocals also have the same number of occurances = 13
sum of these two's = 26
add 9 for ten (10) 1's = 26+9 = 35
so distinct values for x/y = 100-35 = 65.

Excellent approach, but 2/3=4/6=6/9, so there are a total of 14*2=28
fractions that can be simplified. Therefore OA= 63 (A)

You are right the list was so long that I forgot noting down a few while I replied to the answer. I have them in my scratchpad!!

Is there a shorter method?

yup. thats correct. so tedious problem that i was messed up with the numbers....

This is really a gripping question!

We know that all pairs of (x,y) in which x and y have only 1 as their common factor form distinct values for x/y --> consider (x,y) in which x,y have common factor other than 1. They're:
(2,4) , (2,6) ,(2,8) , (2,10) , (3,6), (3,9), (4,6) , (4,8) , (4,10), (5,10) , (6,8) , (6,9), (6,10) , (8,10)
Each of these form 2 indistinct x/y --> 14*2= 28 indistinct x/y

2/2, 3/3, ......10/10 also form indistinct x/y --> 9 such x/y

---> the number of distinct x/y= 100 - 28- 9= 63

kevin,

do you have OA and OE?

I would really like to know how this works...

thanks...