7.4. Little One gave Karlson 111 candies. They immediately ate some of them together, $45\%$ of the remaining candies went to Karlson for lunch, and a third of the candies left after lunch were found by Fräulein Bok during cleaning. How many candies did she find?
Answer: 11.

First method. Let Carlson have had $n$ candies before lunch. Then, after lunch, there were $\frac{55}{100} n$ left, and Fräulein Bock found $\frac{1}{3} \cdot \frac{55}{100} n=\frac{11 n}{60}$ candies. Since the number of candies must be an integer, $11 n$ is divisible by 60. Since the numbers 11 and 60 are coprime, $n$ must be a multiple of 60. There is only one number that is divisible by 60 and less than 111, which is 60. Therefore, $n=60$, meaning Fräulein Bock found 11 candies.

Similar reasoning: if Carlson had $n$ candies before lunch, then he ate $0.45 n=\frac{9 n}{20}$ candies, and since the numbers 9 and 20 are coprime, $n$ is divisible by 20. After lunch, there were $0.55 n=\frac{11 n}{20}$ candies left, and this number must be divisible by 3. But 11 is not divisible by 3, so $n$ must be divisible by 3. Thus, $n$ is divisible by 3 and on