Problem 1006

Answer: It could not. Solution: Suppose the opposite: the products $ab, bc$, and $ca$ end with the two-digit numbers $n, n+1$, and $n+2$, respectively. Among these three consecutive numbers, there must be an odd number, meaning the product of some two of the numbers $a, b$, and $c$ is odd. This implies that at least two of the numbers $a, b$, and $c$ are odd. But then the third number is even, otherwise all three products $ab, bc$, and $ca$ would be odd, which is impossible.

Thus, among the products, one is odd and two are even, meaning the number $n$ is even. Then the number $a$ is even, and the numbers $b$ and $c$ are odd. Now, if $a$ is divisible by 4, then both numbers $n$ and $n+2$ must be divisible by 4. If $a$ is not divisible by 4, then the numbers $n$ and $n+2$ are also not divisible by 4. However, among the two consecutive numbers $n$ and $n+2$, one must be divisible by 4, and the other is not. Contradiction.