If p and q are both positive integers, then is p divisible by 9?

1) p/10 + q/10 is an integer.

2) p/9 + q/10 is an integer.

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(1) p/10 + q/10 is an integer

Let p =8, q = 2, This satisfies our constraint and p is not divisible by 9.

Let p =9 and q = 1. This satisfies our constraint and p is divisible by 9 NS

(2) p/9 +q/10 is an integer.

So, our target denominator for the sum of our terms is going to be 90 (we will have to convert p/9 + q/10 to a common denominator of 90). This results in q being multiplied by 9 and p being multiplied by 10 to convert to the LCD of 90.
Then our q term will contribute one of the following values to the numerator of our sum 9(1), 9(2), 9(3)...= 9,18,27,36,45,54,63,72,81,90....
Our p term will be multiplied by 10 when converting to our common denominator. So, our p term will contribute one of the following values to our final sum:
10,20,30,40,50,60,70,80,90,.... Notice, that our sum will only be divisible by 90, if both 10p and 9q are both multiples of 90, therefore p is a multiple of 9 sufficient.