If the 10- digit number $$897359y7x2$$ is divisible by $$72$$, then what is the value of $$(3x-y)$$ if the value of $$y$$ is the greatest possible?

Given, $$897359y7x2$$ is divisible by 72, the number should be divisible by 8 and 9. If the number is divisible by 8, then the last three digits should be divisible by 8

$$\Rightarrow  7x2$$ is divisible by 8, and thus $$x$$ can be $$1$$, $$5$$, or $$9$$.

If the number is divisible by 9, then the sum of the digits of the number should be divisible by 9

$$\Rightarrow  8 + 9 + 7 + 3 + 5 + 9 + y + 7 + x + 2 = 50+x+y$$ is a multiple of $$9$$.

• For $$x = 1$$
  $$50 + y + 1$$ is a multiple of 9.
  $$\Rightarrow  51 + y $$ is a multiple of 9.
  The only possible value of y is 3.
• For $$x = 5$$
  $$50 + y + 5$$ is a multiple of 9.
  $$\Rightarrow  55 + y$$ is a multiple of 9.
  The only possible value of y is 8.
• For $$x = 9$$
  $$50 + y + 9$$ is a multiple of 9
  $$\Rightarrow  59 + y $$ is a multiple of 9
  The only possible value of y is 4.

So when x = 5, y has the greatest value 8. Therefore $$3x - y = 3(5) - 8 = 7$$ is the correct answer.

Hence, the correct answer is Option C.