### Divisibility Tricks Applied to 693

Some quick divisibility tricks applied to the number 693:

- Every counting number is divisible by 1
- 693 is not even so it isn’t divisible by 2
- Every digit of 693 is divisible by 3, so 693 is divisible by 3
- Since it isn’t divisible by 2, it isn’t divisible by 4
- 693 doesn’t end in a 5 or 0, so it’s not divisible by 5
- 693 is divisible by 3 but not by 2 so it isn’t divisible by 6
**69**– 2(**3**) = 63, a multiple of 7 so 693 is divisible by 7- Since it isn’t divisible by 2 or 4, it can’t be divisible by 8
- 6 + 9 + 3 = 18, a multiple of 9 so 693 is divisible by 9
- Since the last digit of 693 isn’t 0, it is not divisible by 10
**6**–**9**+**3**= 0, so 693 is divisible by 11

693 is a palindrome in several bases:

- 1010110101 BASE 2; note 1(512) + 0(256) + 1(128) + 0(64) + 1(32) + 1(16) + 0(8) + 1(4) + 0(2) + 1(1) = 693
- 3113 BASE 6; note 3(216) + 1(36) + 1(6) + 3(1) = 693
- 414 BASE 13; note 4(169) + 1(13) + 4(1) = 693
- 313 BASE 15; note 3(225) + 1(15) + 3(1) = 693

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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- 693 is a composite number.
- Prime factorization: 693 = 3 x 3 x 7 x 11, which can be written 693 = (3^2) x 7 x 11
- The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 693 has exactly 12 factors.
- Factors of 693: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693
- Factor pairs: 693 = 1 x 693, 3 x 231, 7 x 99, 9 x 77, 11 x 63, or 21 x 33
- Taking the factor pair with the largest square number factor, we get √693 = (√9)(√77) = 3√77 ≈ 26.324893

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Ricardo tweeted his work for this puzzle, too.

@findthefactors 56 and 35 are good hints, help to know where place 5 🙂 pic.twitter.com/kosDETcAeo

— riccardo (@ricca9380) November 27, 2015

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