145. Three friends had a monkey. They bought a certain number of mango fruits and hid them. At night, one of the friends wanted to indulge, went to the storeroom, and wanted to take a third of the fruits, but it turned out that one fruit was left over, and he gave it to the monkey, and ate his share. After some time, the second one came and, not suspecting that the first one had already taken his share, divided the remainder into three equal parts, and again one fruit was left over, and he gave it to the monkey and ate his share. Finally, the third one came and also divided the remainder into three equal parts, and again one fruit was left over, which he gave to the monkey, and ate his share.

The next morning, all three came to the storeroom and divided the remaining fruits equally among themselves, and again one fruit was left over, which they gave to the monkey.

Determine the smallest number of fruits that satisfies the above conditions of distribution.

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## Problem from the tractate "Bava Batra".

Talmud.
145. This problem was proposed by an Indian professor to Delbo.

Solution: Let $x$ be the number of fruits each received in the morning. Then $3x + 1$ is what remained in the morning. The third one took $\frac{1}{2}(3x + 1)$, so he found $\frac{3}{2}(3x + 1) + 1$. The second one took $\frac{1}{2}\left[\frac{3}{2}(3x + 1) + 1\right]$, and found $\frac{3}{2}\left[\frac{3}{2}(3x + 1) + 1\right] + 1$; the first one took $\frac{3}{4}\left[\frac{3}{2}(3x + 1) + 1\right] + \frac{1}{2}$, or $\frac{27x + 19}{8} = 3x + 2 + \frac{3(x + 1)}{8}$, which must be an integer. The smallest value of $x$ that satisfies this condition is 7. Therefore, the first one took 26 fruits and gave 1 to the monkey, the second one took 17 fruits and gave 1 to the monkey. The third one took 11 fruits and gave 1 to the monkey. In the morning, each received 7 fruits, i.e., a total of 21, and one was given to the monkey. The total number of fruits: $26 + 17 + 11 + 21 + 4 = 79$.

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The culture of precise knowledge among the ancient Jews developed under the influence of Babylon and Egypt.

The influence of the former was particularly evident in Jewish metrology. Initial geometric knowledge was of a narrowly practical nature, such as measuring fields, determining the capacity of vessels, etc. Arithmetic knowledge was necessary for keeping household accounts. The main monument of rabbinical writing is the "Talmud" (i.e., teaching, explanation), which was created between 300 BC and 500 AD. This is a kind of huge encyclopedia, where a number of treatises contain a lot of information on astronomy, mathematics, and metrology. A significant number of scholars, translators, and commentators in the field of exact knowledge worked at the courts of caliphs during the later periods of the flourishing of Arab rule, with many Jews becoming Arabized. The Middle Ages also produced several names. In modern times, when mathematical creativity became a common human heritage, Jewish mathematical thought usually merged into the sphere of the nationality with which the Jews were territorially connected. Astronomical and calendar research and works dedicated to the study of ancient written monuments stood apart.