Proof of The Mathematical Fact!

What was that fact?

For a moment, let's assume that such group of 2 numbers exists whose product is equal to sum of rest 13 numbers taken out of 15 numbers.

Let x and y be those numbers in group B. Now x and y can be any number from 1 to 15. 

As per condition,

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 - x - y = xy 

120 = xy + x + y 

Adding 1 to both sides,

121 = xy + x + y + 1

121 = x( y + 1 ) + 1( y + 1 )

121 = ( x + 1 ) ( y + 1 ) 

Since x & y are the numbers in between 1 to 15, possible value of x & y satisfying the above equation is 10. But x & y are must be 2 different number. Hence, our assumption goes wrong here!

So, the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.