Jim found the quotient of two intergers and got a posotive interger. he added another interger to the quotient and got a posotive interger. his sis kim says that all the intergers jim used to get the results must be posotive. do you agree?

I do not agree. On the contrary, I disagree, and I assert that Kim's statement consists largely of horse feathers. I shall illustrate my position by means of a counter-example:

Let the first two numbers be -12 and -3.
Their quotient is +4.
Now, to the quotient, add the NEGATIVE number -1
The result is +3.

All the numbers Jim used to get the results were negative.

And another thing ! There's no such thing as "interger". Whole numbers are "integers".

Geometric pattern of 2 i’m guessing. not really sure about what you’re asking but i hope this !

The answer is biodiversity resulting from new ancestors.