Problem 9.5. Buratino has many coins of 5 and 6 soldi, more than 10 of each type. Coming to the store and buying a book for $N$ soldi, he realized that he could not pay for it without receiving change. What is the greatest value that the natural number $N$ can take if it is no more than 50?
Answer: 19.

Solution. It is easy to check that with $N=19$ change is necessary.

Notice that numbers from 20 to 24 do not meet the condition, as they can be paid for without change: $N=20=4 \cdot 5, 21=3 \cdot 5+6, 22=2 \cdot 5+2 \cdot 6, 23=5+3 \cdot 6, 24=4 \cdot 6$.

It is clear that then numbers from 25 to 50 also do not meet the condition, as each of them is the sum of one or more terms equal to 5, as well as a number from 20 to 24, which is the sum of terms, each equal to 5 and 6.