Question 56
A number is called *nilless* if it is integer and positive and contains no zeros. You can make a positive integer *nilless* by simply omitting the zeros. We denote this with square brackets, for example $[2050] = 25$ and $[13] = 13$. When we multiply, add, and subtract we indicate with square brackets when we omit the zeros. For example, $[4 \cdot 5] + 7 = [20] + 7 = [2 + 7] = [9] = 9$ and $[5 + 5] + 9 = [10] + 9 = [1 + 9] = [10] = 1$. The following is known about the two numbers $a$ and $b$:
* $a$ and $b$ are nilless,
* $1 \lt a \lt b \lt 100$,
* $[[a \cdot b] - 1] = 1$.
Which pairs $(a, b)$ satisfy these three requirements?
* $a$ and $b$ are nilless,
* $1 \lt a \lt b \lt 100$,
* $[[a \cdot b] - 1] = 1$.
Which pairs $(a, b)$ satisfy these three requirements?