## A Multiplication Based Logic Puzzle

### 660 My Two-Year Blogiversary

Today is my 2 year blogiversary and my 660th post. Thank you to all my readers and followers. I really appreciate all of you.

What better way is there to celebrate than with a gorgeous cake? Enjoy!

The cake method of finding prime factors makes a beautiful cake for the number 660 because 660 has several prime factors and the largest one, 11, looks like a couple of candles to top it off perfectly.

660 has a lot of factors, 24 in fact. 660 is a special number for several reasons:

No number less than 660 has more factors than it does, but 360, 420, 480, 504, 540, 600, and 630 each have just as many.

660 has so many factors that it seems natural for it to between twin primes, 659 and 661.

660 is the hypotenuse of the Pythagorean triple 396-528-660. What is the greatest common factor of those three numbers?

660 is the sum of consecutive prime numbers three different ways. Prime number 101 is in two of those ways:

• 157 + 163 + 167 + 173 = 660
• 101 + 103 + 107 + 109 + 113 + 127 = 660
• 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 660

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• 660 is a composite number.
• Prime factorization: 660 = 2 x 2 x 3 x 5 x 11, which can be written 660 = 2² x 3 x 5 x 11
• The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 x 2 = 24. Therefore 660 has exactly 24 factors.
• Factors of 660: 1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330, 660
• Factor pairs: 660 = 1 x 660, 2 x 330, 3 x 220, 4 x 165, 5 x 132, 6 x 110, 10 x 66, 11 x 60, 12 x 55, 15 x 44, 20 x 33, or 22 x 30
• Taking the factor pair with the largest square number factor, we get √660 = (√4)(√165) = 2√165 ≈ 25.690465.

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#### Comments on: "660 My Two-Year Blogiversary" (10)

1. Happy blogiversary!!! 😀 (Your cake is fabulous!)

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• Thank you! I do think it’s the best cake I’ve ever made.

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2. Did you work it out so that you could have your blog’s birthday post on such a fabulous number?
Happy blog birthday, and wishes for many more. xx

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• Thank you for the well wishes.
As far as working out the number I would highlight for this post, it must have been meant to be because I didn’t plan it ahead of time.

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3. This sort of sets my head spinning ‘660 has so many factors that it seems natural for it to between twin primes, 659 and 661″

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4. Congratulations! Happy anniversary!

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• Thank you very much! Also thank you for your continual support!

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5. I kept thinking about 660 being between two primes. I realized that 6 is also flanked by primes, as is 60 (right? hope I’ve got that right.) Thanks for keeping my brain spinning. Yes, I do get dizzy,

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• You are right about all those numbers. I am glad to be of service, and it is a pleasure for me to learn from you as well.

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