## Task A-4.1.

A natural number is called Babylonian if it is greater than 9 and if its representation in base 60 is equal to its decimal representation without the leading digit. For example, the number 123 is Babylonian because $123=(23)_{60}$. How many Babylonian numbers are there less than 10000?
## Solution.

A two-digit number $\overline{a b}$ cannot be Babylonian because it would then be $10 a + b = b$, which means $a = 0$. Therefore, there are no two-digit Babylonian numbers.

A three-digit number $\overline{a b c}$ is Babylonian if and only if $100 a + 10 b + c = 60 b + c$, which is equivalent to $2 a = b$.

This means that all three-digit Babylonian numbers are of the form $\overline{12 c}, \overline{24 c}, \overline{36 c}$, and $\overline{48 c}$, where $c$ is any digit. Therefore, there are 40 three-digit Babylonian numbers.

A four-digit number $\overline{a b c d}$ is Babylonian if and only if $1000 a + 100 b + 10 c + d = 3600 b + 60 c + d$, which is equivalent to $20 a = 70 b + c$.

Since the numbers $20a$ and $70b$ are divisible by 10, we conclude that $c$ must also be divisible by 10, which means $c = 0$.

Now we get that $\overline{a b c d}$ is Babylonian if and only if $2 a = 7 b, c = 0$, and $d$ is any digit.

Since $2 a$ is an even number, $7 b$ must also be even, which means $b$ must be even.

If $b = 0$, then $a = 0$, which is impossible. For $b = 2$, we get $a = 7$, and for $b \geq 4$, we get $a \geq 14$, which is also impossible.

Therefore, all four-digit Babylonian numbers are of the form $\overline{720 d}$, where $d$ is any digit. Thus, there are 10 four-digit Babylonian numbers.

Finally, there are 50 Babylonian numbers less than 10000.