Consecutive numbers will always total an odd number.
Consecutive odd numbers or consecutive primes would be 5 and 7.
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Pick a number. Add or subtract 1 to or from it.
That doesn't work. The number has to be divisible by three. Any three consecutive numbers add up to a multiple of three.
Although not specified as such, "consecutive" requires the numbers to be integers. Two pairs literally means four numbers but there are not four consecutive integers that add up to to 5280. 1318 + 1319 + 1320 + 1321 = 5278 and 1319 + 1320 + 1321 + 1322 = 5282. Four consecutive numbers must add up to an even number that is not a multiple of 4. If by two pairs, the question meant ONE pair (!!), again there is no answer since the sum of any pair of consecutive numbers must be an odd number.
Any pair of consecutive numbers will have an odd total. 10 and 12 are consecutive even numbers that total 22.