A Multiplication Based Logic Puzzle

689 is an amazing number for several reasons. I decided to make graphics to illustrate many of those ways. (689’s factoring information is at the end of this post.)

689 is the sum of consecutive prime numbers 227, 229, and 233.

Also 689 is the sum of the primes from 83 to 109. Do you know what those 7 prime numbers are?

Stetson.edu informs us that 689 is the smallest number that can be expressed as the sum of three different square numbers NINE ways. I decided to figure out what those nine ways are and make this first graphic to share with you:

689 Sum of 3 Different Squares

Note: 614 can also be expressed as the sum of 3 squares 9 different ways, but one of those ways is 17² + 17² + 6² = 614, and that duplicates 17² in the same sum.

689 is the same number when it is turned upside down. Numbers with that characteristic are called Strobogrammatic numbers.

689 Rotation

689 BASE 10 isn’t a palindrome, but 373 BASE 14 is; note that 3(196) + 7(14) + 3(1) = 689

Both of 689’s prime factors have a remainder of 1 when divided by 4, so they are hypotenuses of Pythagorean triples. That fact also means 689 can be expressed as the sum of two square numbers TWO different ways, and it makes 689 the hypotenuse of FOUR Pythagorean triples.  Can you tell by looking at the graphic which two are primitive and which two aren’t?

689 Pythagorean triples

689 is the sum of consecutive numbers three different ways:

  • 344 + 345 = 689; that’s 2 consecutive numbers.
  • 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 = 689; that’s 13 consecutive numbers.
  • 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 689; that’s 26 consecutive numbers.

Now you have a few reasons why 689 is an amazing number. 13 and 53 were part of some of those reasons so it shouldn’t surprise anyone to see 13 and 53 pop up in its factoring information, too:

——————————————————————————

  • 689 is a composite number.
  • Prime factorization: 689 = 13 x 53
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 689 has exactly 4 factors.
  • Factors of 689: 1, 13, 53, 689
  • Factor pairs: 689 = 1 x 689 or 13 x 53
  • 689 has no square factors that allow its square root to be simplified. √689 ≈ 26.248809.

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Comments on: "What Makes 689 Amazing?" (23)

  1. What an interesting number! And here’s one fact you didn’t mention – 689 was the winning number in our jazz club raffle on Saturday. And I – very stupidly – said “Are you sure you’re reading it the right way up?”!

    Liked by 3 people

    • No, no, no. I think you were inspired. How many people learned about the Strobogrammatic property of that number because you asked that? Regardless of how it made you feel, I love your story!

      Like

  2. brandon said:

    And 689 is also the number of votes the current Chief Executive of Hong Kong received out of 1200 selected voters.

    Liked by 2 people

    • That’s interesting. I wonder why only 1200 voters were selected to make such an important choice. 689 is about 57% of 1200.

      Perhaps that is why so far today 628 people from Hong Kong have viewed this blog. That’s so close to 689. I will update today’s view count if it changes. Update: I’ve had 1385 Views from Hong Kong and 1109 views from Taiwan in a single day. Wow.

      Liked by 1 person

      • adrianshum said:

        You pointed out something extremely obvious but neither China nor Hong Kong government is willing to admit/understand

        Like

  3. kevin cheung said:

    689 is the name of the most famous person in hk

    Liked by 1 person

  4. I was brought here because of the “Chief Executive” 689 lol

    Liked by 1 person

  5. MaIngJiu said:

    Magic number for Taiwan and Hong Kong
    Magic!

    Liked by 1 person

  6. Thank you for clarifying it, John Fung!

    Like

  7. Again, 689 for Taiwan!!!

    Liked by 1 person

  8. 689 for Taiwan, Twice!!!

    Liked by 1 person

  9. […] 689 is also a magical number. As findthefactors.com explains, it is a strobogrammatic number, meaning it can be read upside […]

    Like

  10. In the above referenced article the Hong Kong Free Press does an excellent job explaining in English why 689 is causing such a stir in Hong Kong and Taiwan. I recommend reading it.

    Like

  11. […] 689 is also a magical number. As findthefactors.com explains, it is a strobogrammatic number, meaning it can be read upside […]

    Like

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