vrgmat wrote:

Bunuel,

How did you arrive at this? - "Now, this expression to be an integer (1) either b=c=1 must be true or b=c=2 must be true". Why cant b and c be two distinct positive integers such that their decimal add upto 1? How do you concretely say b=c=1 or b=c=2. How do you rule out other possibilities.

I thought ( even though it may not be possible ), we may have numbers like 1/b = 0.6 & 1/c = 0.4 which add upto 1. I thought we may have such decimals which add upto 1. How do you rule out every such possibility?

Even I was thinking on the same lines but later on, I tried to figure it out on my own.

Addressing your query:

Two positive fractions will add up to 1, only when they are in this format : 1/x + (x-1)/x i.e. 1/3 + 2/3 or 1/10 + 9/10.

And x & x-1 are co-prime i.e. there is no common factors between them hence they can never be simplified further.

So, if one of the denominator is Integer, other denominator has to be non-integer to add up to 1.

Or, if both the denominators are integer, sum will never be equal to 1.

Hence, for Statement (2), equation : a/bc= 1/c + 1/b will never be Integer if both c & b will be integer.