## Task 2

In a chain of numbers, each number, after the first, is equal to the product of the digits of the previous number, and the chain continues until a single-digit number is reached. For example: $63 \rightarrow 18 \rightarrow 8$. The chain can consist of two or more numbers. Determine all two-digit numbers that can be at the beginning of a chain that ends with the number 6.
## Solution:

We sequentially write down all possibilities for representing the number 6 as a product of two single-digit numbers, and then we do the same for each two-digit number until we can no longer do so.
![](https://cdn.mathpix.com/cropped/2024_05_30_8f249fc2b96bc1b5a03dg-1.jpg?height=406&width=1512&top_left_y=2170&top_left_x=284)
![](https://cdn.mathpix.com/cropped/2024_05_30_8f249fc2b96bc1b5a03dg-2.jpg?height=452&width=1526&top_left_y=148&top_left_x=207)

There are no other possibilities because the numbers at the beginning of the chain cannot be represented as the product of two single-digit natural numbers, which we can verify by examining the multiplication table up to 10 times 10. Therefore, the number at the beginning of the chain can be one of the following 13 numbers: $16, 23, 28, 32, 44, 47, 48, 61, 68, 74$, 82, 84, and 86.