Not all negative numbers are integers. Some are fractions.
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This is best solved by trial-and-error. If one set of consecutive even integers doesn't work, try a different set. Hint: The integers involved are fairly small.
You can find those by trial and error. You can also write an equation for the three consecutive integers, and solve it. If the first number is "n", the others are "n + 1" and "n + 2". By solving the equation for "n", you get the first of the three numbers.
Oh, dude, okay, so let's break it down. If we have two consecutive odd integers, we can call them x and x+2. So, x + (x+2) = 32. Just solve that simple equation, and you'll find the integers are 15 and 17. Easy peasy, lemon squeezy.
It looks like you are asking how many combinations of 6 numbers are there in the 28 numbers 1 through 28. This is known as the number of combinations of 28 things taken 6 at a time. The answer is 28!/(6!22!) (n! means n factorial, which is the product of all the integers from 1 to n). I get 376,740 if I haven't made an error in arithmetic.