Let $A$ be the set of all numbers which can be represented as the sum of three consecutive positive integers. What is the greatest common divisor of all numbers in $A$?
I need to find a common divisor for all numbers that are the sum of three consecutive positive integers. Let me try some examples and see if I can find a pattern. If the first integer is 1, then the sum is 1 + 2 + 3 = 6. If the first integer is 2, then the sum is 2 + 3 + 4 = 9. If the first integer is 3, then the sum is 3 + 4 + 5 = 12. If the first integer is 4, then the sum is 4 + 5 + 6 = 15. I notice that the sums are all multiples of 3, so 3 is a common divisor. Is 3 the greatest common divisor, or can I find a larger one? I suspect that 3 is the greatest common divisor, because the sums seem to increase by 3 each time. Therefore, 3 is the greatest common divisor of all numbers in $A$. 