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This article mentions an excerpt from A.L. Basham's book A Cultural History of India. The excerpt goes on to give a lot of examples about mathematics in the Vedas. I am mentioning the relevant part

...Of particular interest is the Satapatha Brahmana, which lists all the factors of 720 as far as 24...

If this is true, this indicates a knowledge of divisbility of numbers.

Questions:

Where is such a list mentioned in the Shatapatha Brahmana of the Yajur Veda?

Was this purely a mathematical endeavour? Or, was there some other reason behind this? e.g. in the Sulba Sutras, geometry was studied to do construction for the Yagyas.

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Shatapatha Brahmana gives the divisibility of number 720 which is represented as 360 days + 360 nights. It is used in the case of number and division of bricks and stones which is used in fire altar preparation. Not only this, it also gives information on how microcosm used in fire altar is related with macrocosm. In this way, it also gives information about division of time:

10:4:2:1. Verily, Prajapati, the year, is Agni, and King Soma, the moon. He himself, indeed, proclaimed (taught) his own self to Yagñavakas Râgastambâyana, saying, 'As many lights as there are of mine, so many are my bricks.'

10:4:2:2. Now in this Prajapati, the year, there are seven hundred and twenty days and nights, his lights, (being) those bricks; three hundred and sixty enclosing-stones , and three hundred and sixty bricks with (special) formulas. This Prajâpati, the year, has created all existing things, both what breathes and the breathless, both gods and men. Having created all existing things, he felt like one emptied out, and was afraid of death.

10:4:2:3. He bethought himself, 'How can I get these beings back into my body? how can I put them back into my body? how can I be again the body of all these beings?'

10:4:2:4. He divided his body into two; there were three hundred and sixty bricks in the one, and as many in the other: he did trot succeed 2.

10:4:2:5. He made himself three bodies,--in each of them there were three eighties of bricks I he did not succeed.

10:4:2:6. He made himself four bodies of a hundred and eighty bricks each: he did not succeed.

10:4:2:7. He made himself five bodies,--in each of them there were a hundred and forty-four bricks: he did not succeed.

10:4:2:8. He made himself six bodies of a hundred and twenty bricks each: he did not succeed. He did not develop himself sevenfold 3.

10:4:2:9. He made himself eight bodies of ninety bricks each: he did not succeed.

10:4:2:10. He made himself nine bodies of eighty bricks each: he did not succeed.

10:4:2:11. He made himself ten bodies of seventy-two bricks each: he did not succeed. He did not develop elevenfold.

10:4:2:12. He made himself twelve bodies of sixty bricks each: he did not succeed. He did not develop either thirteenfold or fourteenfold.

10:4:2:13. He made himself fifteen bodies of forty-eight bricks each: he did not succeed.

10:4:2:14. He made himself sixteen bodies of forty-five bricks each: he did not succeed. He did not develop seventeenfold.

10:4:2:15. He made himself eighteen bodies of forty bricks each: he did not succeed. He did not develop nineteenfold.

10:4:2:16. He made himself twenty bodies of thirty-six bricks each: he did not succeed. He did not develop either twenty-one-fold, or twenty-two-fold, or twenty-three-fold.

10:4:2:17. He made himself twenty-four bodies of thirty bricks each. There he stopped, at the fifteenth; and because he stopped at the fifteenth arrangement 1 there are fifteen forms of the waxing, and fifteen of the waning (moon).

10:4:2:18. And because he made himself twenty-four bodies, therefore the year consists of twenty-four half-months. With these twenty-four bodies of thirty bricks each he had not developed (sufficiently). He saw the fifteen parts of the day, the muhûrtas.

Thus as we can see above number 720 which is combination of 360 days + 360 nights is divided as:

720 = 360×2
720=240×3
720=180×4
720=144×5
720=120×6
720=90×8
720=80×9
720=72×10
720=60×12
720=48×15
720=45×16
720=40×18
720=36×20
720=30×24

Thus as division of 720 is stopped at 30×24. Thus there are half of 24 months (as 720 is day+night of year so half is done here) in a year. Thus there are 15 muhurtas in a day and 15 in night thus total 30. As division stops in 15th division thus 15 waxing and 15 waning days in a month. Thus there are 30 days in a month.

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    Awesome! You mean Prajapati? Also, the first Shloka is quite unclear.. – Amit Saxena May 16 '16 at 7:55
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    @AmitSaxena yes, it is referring to Prajapati... – Tejaswee May 16 '16 at 7:56
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    This might be related to the division of year into 720 Upanakshatras as mentioned by Dr David Frawley here: vedanet.com/2012/06/13/nakshatras-and-upanakshatras – Amit Saxena May 16 '16 at 8:22
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    @AmitSaxena Thanks for that link.. There are many interesting concepts there... – Tejaswee May 16 '16 at 8:28
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    Mathematically, it is interesting to notice how one periodic process (namely the year) is associated with another periodic process (cycles of the Moon). This is connected together by the abstract notion of divisibility. – Amit Saxena May 16 '16 at 8:31

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