Question

There are 13 fractions, whose numerators and denominators are 1, 2, ..., 26, each appearing
exactly once either as a numerator or a denominator. At most how many of these fractions can
be simplified to integers? ...Please answer my question ​

Answer 1

Step-by-step explanation:

By arranging fractions this way, we get

26/13  = 2

25/5   = 5

24/4    = 6

23/1 = 23

22/11   = 2

21/7   = 3

20/10   = 2

19/17  

18/9   = 2

16/8     = 2

15/3   = 5

14/2   = 7

12/6  = 2

So, it is clear that by this arrangement, all of them can be converted to integers except from one e.g.

19/17

Answer 2

Given :   There are 13 fractions, whose numerators and denominators are 1, 2, ..., 26, each appearing  exactly once either as a numerator or a denominator.  

To find : At most how many of these fractions can  be simplified to integers

Solution:

There are 26 Numbers  

We need to get Maximum integers

Let see prime number  2  ,  3  ,   5  , 7   ,  11   ,   13   ,  17  ,  19  ,  23  

we can use prime upto 13  as we their multiplier as 2 in range upto 26

now we left with 17 , 19 , 23

we can use one using 1 in Denominator

so we left with 2    so we can make fraction of that

Hence atleast one fraction will be there which is not integer

so atmost 12 fractions  can  be simplified to integers

One of the Example :

26/13  ( Integer)

25/5   ( Integer)

24/4     ( Integer)

23/1   ( Integer)

22/11   ( Integer)

21/7   ( Integer)

20/10   ( Integer)

19/17    (not Integer)

18/9     ( Integer)

16/8     ( Integer)

15/3     ( Integer)

14/2     ( Integer)

12/6      ( Integer)

Hence atmost 12  fractions can  be simplified to integers

atmost 12  fractions can  be simplified to integers

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