SOLUTION: The expression x^2 -x -a, where 0 < a < 100 can be factored into (x-p)(x+q), where p,q are positive integers. How many different values of a are there?

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Question 1207397: The expression x^2 -x -a, where 0 < a < 100 can be factored into (x-p)(x+q), where p,q are positive integers. How many different values of a are there?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


%28x-p%29%28x%2Bq%29=x%5E2%2B%28p-q%29x-pq

The requirements are that (1) p and q are positive integers; (2) p-q = -1, or p = q-1; and (3) pq is less than 100.

Simple enumeration shows that p can have any integer value from 1 to 9 inclusive. That's 9 values.

ANSWER: 9

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

I will re-write the solution by @greenestamps, correcting mistakes there.


%28x-p%29%28x%2Bq%29=x%5E2%2B%28p-q%29x-pq

The requirements are that (1) p and q are positive integers; (2) -p+q = -1, or p = q+1; and (3) pq is less than 100.

Simple enumeration shows that p can have any integer value from 2 to 10 inclusive. That's 9 values.

ANSWER: 9

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Now everything is correct.