Find the Lowest Whole Solution

Taking,the equation modulo 3,we get: $3x+13y\equiv y\equiv 0 \mod 3$ Which means that $y=3y_1$ for some positive integer $y_1$ .

Now taking the equation modulo 13,we get: $3x+13y\equiv 3x\equiv 0 \mod 13\rightarrow x\equiv 0 \mod 13$ Which tells us that $x=13x_1$ for some positive integer $x_1$ .Now the given equation becomes: $3x+13y = 234 \\ 3(13x_1)+13(3y_1) = 234 \\ x_1+y_1 = \dfrac{234}{39}\\ x_1+y_1 = 6$ We want to minimize $y$ .To minimize $y$ ,we have to find the smallest positive integer $y_1$ which satisfies the equation.Obviously, $y_1=1$ is the smallest possible integer value of $y_1$ ,hence $y=3y_1=3(1)=\boxed{3}$ is the smallest possible value of $y$

3x +13y = 234

x = (234 – 13y)/3

= 234/3 – 13y/3

to have integer value 13y must be divisible by 3

so minimum value of y = 3

let see 3x, 234mod3=0 y can be 0，then x should be 78. but y should be positive the y can only be the multiple of 3. so the lowest ans should be 3.

nice solution