A Vort for the Mathematicians
עזים מאתים ותישים עשרים רחלים מאתים ואילים עשרים
“200 she-goats and 20 he-goats; 200 ewes and 20 rams” (Bereishis 32:15)
Prior to his encounter with his enraged brother Eisav, Yaakov sent many animals as gifts in an effort to placate him. Specifically, the Torah says that Yaakov sent 200 female goats and 20 male goats, 200 ewes and 20 rams, 30 nursing camels with their young, 40 female cows and 10 bulls, and 20 female donkeys and 10 male donkeys. Since there are no coincidences in the Torah, Yaakov must have had a calculated reason for the number of animals that he sent.
Rav Nachshon Gaon, the ninth-century leader of the yeshiva in Sura, provides a brilliant explanation for the number of goats that Yaakov sent, which totaled 220. He points out that the proper factors of 220 (numbers less than 220 that evenly divide into it) are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. These numbers add up to 284. Similarly, the proper factors of 284 are 1, 2, 4, 71, and 142, which add up to 220. The numbers 220 and 284 are unique in that the sum of the proper factors of each adds up to the other number, and they are the smallest set of numbers with this unusual characteristic.
A pair of numbers with this rare property is referred to by mathematicians as amicable numbers, or as Rav Nachshon Gaon calls them, מנין נאהב – beloved numbers. He writes that there was an ancient custom that when a person wanted to curry favor with a prominent individual, he would present him with items numbering one of a pair of amicable numbers and keep the corresponding number for himself. He suggests that this is what Yaakov did to pacify Eisav, sending him 220 goats while keeping 284 for himself. It is clear from the text that Yaakov sent 200 female goats and 20 male goats, for a total of 220 goats, but where do we see that he kept 284 for himself?
When Yaakov instructed his servants about giving the presents to Eisav, he told them: אכפרה פניו במנחה ההולכת לפני –” I will appease him (Eisav) with the gifts that precede me” (32:21). The word אכפרה can be split into two words: אך פרה. The term אך is used to connote limiting or reducing (see Yerushalmi Berachos 67b). In this case, the numerical value of פרה is 285, so diminishing it by 1 yields 284, which is the number of goats Yaakov kept for himself as part of his attempt to pacify Eisav. (R’ Ozer Alport)