A number theory problem by Emmanuel Ngonyama

Number Theory Level 2

$X$ and $Y$ are positive integers such that $X + Y + XY = 54$ , What is the numerical value of $X+Y$ ?

1 solution

Emmanuel Ngonyama
Jan 22, 2016

First add one on both sides

x+y+xy+1=55 xy+x+y+1=55

factorize both side (prime factorization on the right side)

(x+1)(y+1)=(11)(5)

Since x and y are positive integers therefore x+1 and y+1 are also positive integers

x+1+y+1=11+5

x+y+2=16

x+y=14

note: (x+1)(y+1)=(55)(1) is incorrect since x or y will be equal to 0 (x and y are positive)

I think your method is best suited for objective .

I think this will be more clear and descriptive :

$(x + 1)(y + 1) = (11)(5)$

$x + 1 = 11 \implies x = 10$

$y + 1 = 5 \implies y = 4$

$So, x + y = 10 + 4 = 14$

This is just my method .

Ram Mohith - 3 years, 1 month ago