If x and y are positive integers, each greater than 1, and if 13(x - 1) = 17(y - 1), what is the least possible value of (x + y)?

Solution:

13(x - 1) = 17(y - 1) .....(1)

⇒ 13x - 13 = 17y - 17

⇒ 13x + 4 = 17y

⇒ 4 = 17y - 13x

⇒ 4 - 4y = 13y - 13x

⇒ 4(1 - y) = 13(y - x)

Since x and y are integers, we know that (y - x) is an integer, which means 13(y - x) is a multiple of 13.

From this, we can conclude that 4(1 - y) is a multiple of 13

What is the smallest value of y (given that y is a positive integer greater than 1) such that 4(1 - y) is a multiple of 13?

If y = 14, then 4(1 - y) = 4(1 - 14) = 4(- 13) = - 52

So, y = 14 is the smallest value of y to meet the given conditions.

Now, putting the value of y = 14 in equation (1) we get,

13(x - 1) = 17(y - 1)

⇒ 13x - 13 = 17(14 - 1)

⇒ 13x = 17 × 13 + 13

⇒ 13x = 18 × 13

∴ x = 18

∴ x + y = 18 + 14 = 32