Triples

A brainteaser from the Soviet science magazine Kvant, via Quantum, January/February 1991:

Bobby found the sum of three consecutive integers, then of the next three consecutive integers, then multiplied these two sums together. Could the product have been 111,111,111?

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No. One of the two groups must contain one even number and two odd ones. The sum of these three numbers must be even, so the product of the two sums must be even as well.

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