[ GCD and LCM. Mutual simplicity ]
[ Divisibility of numbers. General properties ]

Authors: Dritriev O. Zhennodarov R.G.

Petya chose 10 consecutive natural numbers and wrote each one either in red or blue pencil (both colors are present).

Can the sum of the least common multiple of all red numbers and the least common multiple of all blue numbers end in 2016?
Note that a number ending in 2016 is divisible by 16.

Among the ten of Petya's numbers, there are either one or two numbers divisible by 8. In the first case, one of the obtained least common multiples (LCM) is divisible by 8, and the other is not, so their sum is not divisible even by 8. In the second case, the difference between the two of Petya's numbers divisible by 8 is 8, so one of them is divisible by 16, and the other is not. Therefore, one of the LCMs is divisible by 16, and the other is not. Thus, in this case, the sum of the LCMs cannot be divisible by 16 either.

## Answer

It cannot.