Friday, December 1, 2017

Goats and Amicable Numbers

Goats and Amicable Numbers

In this week’s parasha, we find Yaakov preparing for his encounter with his twin brother Esav in several ways. Among other preparations, Yaakov sends him gifts consisting of various different kinds of animals. The Torah tells us (Bereishit 32:14–16) how many of each kind of animal Yaakov sent: 200 female goats and 20 male goats; 200 female sheep (ewes) and 20 male sheep (rams); 30 nursing camels with their young; 40 female cows and 10 bulls; 20 female donkeys and 10 male donkeys. What is the significance of these numbers?

In his ספר בעלי ברית אברם, R’ Avraham Azulai provides an explanation for the number of goats, which he attributes to R’ Nachshon Gaon of the 9th century. The total number of goats is 200 + 20 = 220. What significant property does the number 220 have?

Consider the factors of 220, that is, numbers that multiply together to give the product 220. We can factor the number 220 in the following ways:
220 = 1 × 220
220 = 2 × 110
220 = 4 × 55
220 = 5 × 44
220 = 10 × 22
220 = 11 × 20
Now consider only the “proper factors” of 220 – that is, all the factors in the above list, excluding the number 220 itself – and add them up:
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
So the proper factors of 220 add up to 284.
We now repeat the process, considering the factors of 284. We can factor the number 284 in the following ways:
284 = 1 × 284
284 = 2 × 142
284 = 4 × 71
Again, we consider only the proper factors of 284 – all the factors in the above list, excluding the number 284 itself – and add them up:
1 + 2 + 4 + 71 + 142 = 220
So the proper factors of 284 add up to 220. Does this number look familiar?

As we have just shown, the numbers 220 and 284 have the property that the proper factors of each number add up to the other number. A pair of numbers with this property is known as a pair of amicable numbers, or according to R’ Nachshon Gaon, מנין נאהב. Apparently it was known to the ancients that in order to gain the love of kings and princes, a person would give one of a pair of amicable numbers as a present, keeping the other number for himself. This is so that the factors of the number given add up to the number kept, and the factors of the number kept add up to the number given. So this is what Yaakov did. He sent Esav 220 goats, and kept 284 for himself.

Wait a minute: The Torah tells us that Yaakov gave Esav 220 goats, but where do we see in the Torah that he kept 284 for himself? Several pesukim later, as Yaakov gives instructions to the servants carrying the gifts, the Torah records (32:21), “כי־אמר אכפרה פניו במנחה ההולכת לפני” – “for he said: I will win him over with the gifts that are being sent ahead.” R’ Nachshon Gaon explains that this sentence contains a hint to the number 284, in the following way. The word אכפרה can be divided in two parts: אכ פרה. When the Torah uses the word אך, it is generally interpreted by the Rabbis to indicate exclusion or reduction. Calculating the numerical value of the second part of the word, פרה, we get: 80 (פ‎) + 200 (ר‎) + 5 (ה‎) = 285. Applying a reduction (indicated by אך) to the value 285 (given by פרה), we obtain a value of 284. This represents the number of goats that Yaakov kept for himself, according to R’ Nachshon Gaon.

Special thanks to Daniel Levenstein for bringing this insight to my attention.


Leonard Eugene Dickson, History of the Theory of Numbers, Volume I: Divisibility and Primality, Carnegie Institute of Washington: Washington, 1919, p. 39, available at:

ר' אברהם ב"ר מרדכי אזולאי, ספר בעלי ברית אברם, published 1873 but existed in manuscript for 300 years previously; pp. 48–49, available beginning at:

הנאהבים והנעימים - על רעות אצל מספרים, in Michlalah Jerusalem College's mathematical journal אלף אפס (ℵ₀):
(may not work in all browsers)
(Thanks to Yaaqov Loewinger for this link via Hebrew Wikipedia)

Thursday, January 5, 2017

Can you Count to 70?

Question: How many males are counted as coming to Mitzrayim with Yaakov? One thing is for sure, it wasn't 70. I still have not been able to figure out how all the numbers worked - who were the 66 mentioned in 46:26 and the 70 in 46:27? 66+3 = 69, the last time I checked. If you add up all of the children and grandchildren, it does come out to 70 but then it should have been 67 and then 70. All that aside, it was not only males who were counted. Dinah is counted along with her brothers which is understandable. Serach bas Asher is counted as well which is slightly more puzzling. One must assume she was not the only granddaughter. From Rashi (46:27) it seems Yocheved was somehow part of the 70 as well.

While I was not able to find anything explaining why these particular women figured in the count as opposed to others, I did see an interesting insight into the pesukim in consideration of that fact. Tzeror HaMor and Emes L'Yaakov both point out a discrepency in the per-wife tallies found in the pesukim. The numbers for Rachel ("arba'ah asar") and Bilhah ("shiv'ah") are of the masculin form. The numbers for Leah ("sheloshim veshalosh") and Zilpah ("sheish esreih") are feminine. They both explain that Leah and Zilpah both had women counted among their offspring - Dinah from Leah and Serach from Zilpah. Therefore their numbers are delivered in feminine. Rachel and Bilhah had no feminine offspring counted and thus their numbers are in masculine.

One might wonder why this is so, considering that the generic plural is usually masculine by default. However, Emes L'Yaakov points out that the word "nefesh" which the number is qualifying is feminine. So the default gender of the number for "nefesh" should be feminine. Rachel and Bilhah were the exceptions.