What five consecutive integers have the sum of 265?

The numbers are 51, 52, 53, 54 and 55.

You can find this easily because the average (mean) of the 5 numbers will

be the sum (265) divided by 5. (which is twice the sum divided by 10, 530/10 = 53)

265/5 = 53, which becomes the center number, and the two numbers to either side, 53-2, 53-1, 53+1, 53+2 are the other four.

Another example:

the 5 consecutive numbers that total to 1205 are 239, 240, 241, 242, 243

because 1205/5 = 2410/10 = 241. As long as the sum ends in 5 or zero, it is straightforward.