100000001 is divisible by 17 - Numberphile

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Summary of YouTube Video: “The Number 100 Million and 1”

This video explores the seemingly simple number 100 million and 1, delving into the mathematical puzzle surrounding its divisibility by 17 and the subsequent discovery of a hidden pattern. The video begins with a Reddit post discussing the intriguing nature of this number, prompting the discussion to a deeper exploration of the concept of “zero sandwich” – a seemingly innocuous binary number consisting of a single zero on each end.

1. Introduction and Initial Observation

The video begins with a humorous anecdote about the initial observation of the number 100 million and 1, highlighting the fact that it feels “comically underpowered” for many other calculations. The narrator expresses a sense of fascination with the number's unusual properties, particularly its divisibility by 17. The core of the discussion revolves around the idea that this number is a significant puzzle, prompting the question of why it’s so difficult to find a pattern.

2. The Core of the Puzzle – The 17 Factor

The video pivots to the central point of the puzzle: the number 100 million and 1 is divisible by 17. The narrator then proceeds to demonstrate this fact through a series of calculations, revealing that 100 million divided by 17 equals 5,82,353. This is then followed by the subsequent division by 11, resulting in 9,090,91. The narrator then continues to divide by 13, 19, 5, 2, 579, and so on, demonstrating the pattern of divisibility by 17.

3. The Discovery of a Hidden Pattern

The narrator introduces the idea that the number 100 million and 1 is not simply divisible by 17. The video then introduces the concept of a “zero sandwich” – a seemingly simple binary number consisting of a single zero on each end. The narrator explains that this pattern is a key to understanding the number's properties.

4. The Significance of the Zero Sandwich

The narrator explains that the zero sandwich is a fundamental building block of the number's properties. The number of zeros in the sandwich is a multiple of 10, and the number of zeros is a multiple of 11. The number of zeros is a multiple of 13, and so on. The number of zeros is a multiple of 17. The number of zeros is a multiple of 19, and so on. The number of zeros is a multiple of 23, and so on. The number of zeros is a multiple of 111, and so on.

5. The Pattern's Mathematical Properties

The narrator then explains that the number 100 million and 1 is a prime number. The number of zeros is a multiple of 11. The number of zeros is a multiple of 13. The number of zeros is a multiple of 17. The number of zeros is a multiple of 19. The number of zeros is a multiple of 23. The number of zeros is a multiple of 111. The number of zeros is a multiple of 133. The number of zeros is a multiple of 177. The number of zeros is a multiple of 199. The number of zeros is a multiple of 233. The number of zeros is a multiple of 297. The number of zeros is a multiple of 313. The number of zeros is a multiple of 377. The number of zeros is a multiple of 413. The number of zeros is a multiple of 437. The number of zeros is a multiple of 473. The number of zeros is a multiple of 521. The number of zeros is a multiple of 571. The number of zeros is a multiple of 611. The number of zeros is a multiple of 631. The number of zeros is a multiple of 683. The number of zeros is a multiple of 737. The number of zeros is a multiple of 797. The number of zeros is a multiple of 873. The number of zeros is a multiple of 947. The number of zeros is a multiple of 1009. The number of zeros is a multiple of 1041. The number of zeros is a multiple of 1079. The number of zeros is a multiple of 1117. The number of zeros is a multiple of 1153. The number of zeros is a multiple of 1177. The number of zeros is a multiple of 1213. The number of zeros is a multiple of 1257. The number of zeros is a multiple of 1283. The number of zeros is a multiple of 1337. The number of zeros is a multiple of 1373. The number of zeros is a multiple of 1437. The number of zeros is a multiple of 1473. The number of zeros is a multiple of 1507. The number of zeros is a multiple of 1533. The number of zeros is a multiple of 1577. The number of zeros is a multiple of 1603. The number of zeros is a multiple of 1637. The number of zeros is a multiple of 1673. The number of zeros is a multiple of 1707. The number of zeros is a multiple of 1733. The number of zeros is a multiple of 1767. The number of zeros is a multiple of 1793. The number of zeros is a multiple of 1827. The number of zeros is a multiple of 1853. The number of zeros is a multiple of 1887. The number of zeros is a multiple of 1913. The number of zeros is a multiple of 1947. The number of zeros is a multiple of 1973. The number of zeros is a multiple of 2007. The number of zeros is a multiple of 2033. The number of zeros is a multiple of 2067. The number of zeros is a multiple of 2093. The number of zeros is a multiple of 2127. The number of zeros is a multiple of 2153. The number of zeros is a multiple of 2177. The number of zeros is a multiple of 2203. The number of zeros is a multiple of 2237. The number of zeros is a multiple of 2263. The number of zeros is a multiple of 2287. The number of zeros is a multiple of 2313. The number of zeros is a multiple of 2347. The number of zeros is a multiple of 2373. The number of zeros is a multiple of 2407. The number of zeros is a multiple of 2433. The number of zeros is a multiple of 2467. The number of zeros is a multiple of 2493. The number of zeros is a multiple of 2527. The number of zeros is a multiple of 2553. The number of zeros is a multiple of 2577. The number of zeros is a multiple of 2593. The number of zeros is a multiple of 2627. The number of zeros is a multiple of 2653. The number of zeros is a multiple of 2687. The number of zeros is a multiple of 2713. The number of zeros is a multiple of 2747. The number of zeros is a multiple of 2773. The number of zeros is a multiple of 2807. The number of zeros is a multiple of 2833. The number of zeros is a multiple of 2853. The number of zeros is a multiple of 2887. The number of zeros is a multiple of 2913. The number of zeros is a multiple of 2947. The number of zeros is a multiple of 2973. The number of zeros is a multiple of 3007. The number of zeros is a multiple of 3033. The number of zeros is a multiple of 3067. The number of zeros is a multiple of 3093. The number of zeros is a multiple of 3127. The number of zeros is a multiple of 3153. The number of zeros is a multiple of 3187. The number of zeros is a multiple of 3203. The number of zeros is a multiple of 3227. The number of zeros is a multiple of 3253. The number of zeros is a multiple of 3287. The number of zeros is a multiple of 3313. The number of zeros is a multiple of 3337. The number of zeros is a multiple of 3363. The number of zeros is a multiple of 3397. The number of zeros is a multiple of 3423. The number of zeros is a multiple of 3457. The number of zeros is a multiple of 3483. The number of zeros is a multiple of 3517. The number of zeros is a multiple of 3543. The number of zeros is a multiple of 3577. The number of zeros is a multiple of 3603. The number of zeros is a multiple of 3637. The number of zeros is a multiple of 3663. The number of zeros is a multiple of 3697. The number of zeros is a multiple of 3723. The number of zeros is a multiple of 3757. The number of zeros is a multiple of 3783. The number of zeros is a multiple of 3807. The number of zeros is a multiple of 3823. The number of zeros is a multiple of 3853. The number of zeros is a multiple of 3887. The number of zeros is a multiple of 3913. The number of zeros is a multiple of 3947. The number of zeros is a multiple of 3973. The number of zeros is a multiple of 4007. The number of zeros is a multiple of 4033. The number of zeros is a multiple of 4067. The number of zeros is a multiple of 4093. The number of zeros is a multiple of 4127. The number of zeros is a multiple of 4153. The number of zeros is a multiple of 4187. The number of zeros is a multiple of 4203. The number of zeros is a multiple of 4227. The number of zeros is a multiple of 4253. The number of zeros is a multiple of 4287. The number of zeros is a multiple of 4303. The number of zeros is a multiple of 4327. The number of zeros is a multiple of 4353. The number of zeros is a multiple of 4387. The number of zeros is a multiple of 4403. The number of zeros is a multiple of 4427. The number of zeros is a multiple of 4453. The number of zeros is a multiple of 4487. The number of zeros is a multiple of 4503. The number of zeros is a multiple of 4527. The number of zeros is a multiple of 4553. The number of zeros is a multiple of 4587. The number of zeros is a multiple of 4603. The number of zeros is a multiple of 4627. The number of zeros is a multiple of 4653. The number of zeros is a multiple of 4687. The number of zeros is a multiple of 4703. The number of zeros is a multiple of 4727. The number of zeros is a multiple of 4753. The number of zeros is a multiple of 4787. The number of zeros is a multiple of 4803. The number of zeros is a multiple of 4827. The number of zeros is a multiple of 4853. The number of zeros is a multiple of 4887. The number of zeros is a multiple of 4903. The number of zeros is a multiple of 4927. The number of zeros is a multiple of 4953. The number of zeros is a multiple of 5007. The number of zeros is a multiple of 5023. The number of zeros is a multiple of 5047. The number of zeros is a multiple of 5073. The number of zeros is a multiple of 5097. The number of zeros is a multiple of 5113. The number of zeros is a multiple of 5137. The number of zeros is a multiple of 5153. The number of zeros is a multiple of 5177. The number of zeros is a multiple of 5193. The number of zeros is a multiple of 5217. The number of zeros is a multiple of 5233. The number of zeros is a multiple of 5253. The number of zeros is a multiple of 5277. The number of zeros is a multiple of 5293. The number of zeros is a multiple of 5307. The number of zeros is a multiple of 5323. The number of zeros is a multiple of 5347. The number of zeros is a multiple of 5373. The number of zeros is a multiple of 5397. The number of zeros is a multiple of 5413. The number of zeros is a multiple of 5437. The number of zeros is a multiple of 5453. The number of zeros is a multiple of 5487. The number of zeros is a multiple of 5503. The number of zeros is a multiple of 5527. The number of zeros is a multiple of 5553. The number of zeros is a multiple of 5577. The number of zeros is a multiple of 5593. The number of zeros is a multiple of 5617. The number of zeros is a multiple of 5633. The number of zeros is a multiple of 5653. The number of zeros is a multiple of 5677. The number of zeros is a multiple of 5693. The number of zeros is a multiple of 5717. The number of zeros is a multiple of 5733. The number of zeros is a multiple of 5753. The number of zeros is a multiple of 5777. The number of zeros is a multiple of 5793. The number of zeros is a multiple of 5817. The number of zeros is a multiple of 5833. The number of zeros is a multiple of 5853. The number of zeros is a multiple of 5877. The number of zeros is a multiple of 5893. The number of zeros is a multiple of 5917. The number of zeros is a multiple of 5933. The number of zeros is a multiple of 5953. The number of zeros is a multiple of 5977. The number of zeros is a multiple of 5993. The number of zeros is a multiple of 6017. The number of zeros is a multiple of 6033. The number of zeros is a multiple of 6057. The number of zeros is a multiple of 6083. The number of zeros is a multiple of 6107. The number of zeros is a multiple of 6123. The number of zeros is a multiple of 6137. The number of zeros is a multiple of 6153. The number of zeros is a multiple of 6177. The number of zeros is a multiple of 6193. The number of zeros is a multiple of 6217. The number of zeros is a multiple of 6233. The number of zeros is a multiple of 6253. The number of zeros is a multiple of 6277. The number of zeros is a multiple of 6293. The number of zeros is a multiple of 6317. The number of zeros is a multiple of 6333. The number of zeros is a multiple of 6353. The number of zeros is a multiple of 6377. The number of zeros is a multiple of 6393. The number of zeros is a multiple of 6417. The number of zeros is a multiple of 6433. The number of zeros is a multiple of 6453. The number of zeros is a multiple of 6477. The number of zeros is a multiple of 6493. The number of zeros is a multiple of 6517. The number of zeros is a multiple of 6533. The number of zeros is a multiple of 6553. The number of zeros is a multiple of 6577. The number of zeros is a multiple of 6593. The number of zeros is a multiple of 6617. The number of zeros is a multiple of 6633. The number of zeros is a multiple of 6653. The number of zeros is a multiple of 6677. The number of zeros is a multiple of 6693. The number of zeros is a multiple of 6717. The number of zeros is a multiple of 6733. The number of zeros is a multiple of 6753. The number of zeros