## A Multiplication Based Logic Puzzle

### 480 The Very Inspiring Blogger Award

Nikita Rath recently nominated me for the Very Inspiring Blogger Award. I have really enjoyed reading her blog about her travel adventures across the world. She was born in India and has been to several other very interesting places. I especially loved reading about her trip to Budapest. I was also quite thrilled to read that her favorite school subject was mathematics.

I will write a few random facts about the number 480 before I list my nominations for the Very Inspiring Blogger Award.

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480 is the sum of consecutive primes two different ways.

• 239 + 241 = 480; those 2 consecutive primes also happen to be twin primes.
• 109 + 113 + 127 + 131 = 480; that’s 4 consecutive primes.

No counting number less than 480 has more factors than 480 has, but 360 and 420 each have just as many.

Although I could make a forest of the many different 480 factors trees, I will only include one of the MANY possible trees here:

Since 24 x 20 is one of its factor pairs, and the difference between those two numbers is 4, the next perfect square is only 4 numbers away. The next perfect square is 484 which is 22 x 22. (This fact is a natural consequence of the fact that 2 + 2 = 4 and 2 x 2 = 4. Only numbers that are 4 less than a perfect square can claim a factor pair with a difference of 4.)

480 is the hypotenuse of the Pythagorean triple 288-384-480.

480 is the longer leg of the primitive Pythagorean triple 31-480-481. Since 480 has so many factors, it has too many other Pythagorean triples to list here.

One of my readers gave another very interesting fact about the number 480 in the comments. Check it out!

• 480 is a composite number.
• Prime factorization: 480 = 2 x 2 x 2 x 2 x 2 x 3 x 5, which can be written 480 = (2^5) x 3 x 5
• The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 x 2 x 2 = 24. Therefore 480 has exactly 24 factors.
• Factors of 480: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, 480
• Factor pairs: 480 = 1 x 480, 2 x 240, 3 x 160, 4 x 120, 5 x 96, 6 x 80, 8 x 60, 10 x 48, 12 x 40, 15 x 32, 16 x 30, or 20 x 24
• Taking the factor pair with the largest square number factor, we get √480 = (√16)(√30) = 4√30 ≈ 21.9089

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Now I’ll nominate some very inspiring blog posts for this award:

1. Even though Homeschoolpdx isn’t as comfortable with mathematics as she’d like to be, she has found some very good ways to teach her young children mathematical concepts using storybooks, games, cooking, gardening, and toys! Her children are already very good at math, and I am confident that they will continue to be.
2. Life Through a Mathematician’s Eyes posted favorites for the month for April: favorite mathematical quote, favorite art and maths inspiration, favorite number, favorite mathematician, and favorite blog/pages/people. This blog is well written and quite pleasing to the eye and was the host of the 121st edition of Carnival of Mathematics.
3. Lisa M. Peek wrote a post with a very intriguing title: 3-reasons-that-blog-posts-with-numbers-are-popular. She has noticed that blog posts with lists often go viral on facebook. She has given some thought about why that happens and gives some compelling reasons.
4. Mopdog did the A through Z challenge on 26 Ways to Die in Medieval Hungary. I loved reading every single post. These are stories familiar to every Hungarian but are generally unknown to the rest of the world until these posts were written. A: by Adultery is the best place to start.
5. Remember how fun it was to use your thumb and a flip book to make a cartoon character dance? Paula Beardell Krieg has spent months planning and preparing four wonderful flip books that teach and reach students who are learning to graph linear equations. She has even prepared pdf’s of the pages that can be downloaded and assembled. Complete instructions are given in the-animated-equation-book.

#### Comments on: "480 The Very Inspiring Blogger Award" (2)

1. Thank you Iva for your kind words 🙂 I love solving your puzzles too and thank you for taking time and moving forward the nominations 🙂

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2. abyssbrain said:

Congratulations!

Btw, just a few days ago, I saw these on a number theory book.

(n^8) – 1 is divisible by 480if n is prime and n > 5.

(n^9) – 1 is divisible by 480if n is odd and n > 1

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