538 and Level 2

538 is the hypotenuse of the Pythagorean triple 138-520-538. Can you find the greatest common factor of those three numbers?

538 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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  • 538 is a composite number.
  • Prime factorization: 538 = 2 x 269
  • 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 538 has exactly 4 factors.
  • Factors of 538: 1, 2, 269, 538
  • Factor pairs: 538 = 1 x 538 or 2 x 269
  • 538 has no square factors that allow its square root to be simplified. √538 ≈ 23.194827

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538 Factors

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537 and Level 1

537 is made from 3 consecutive odd numbers so it is divisible by 3.

537 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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  • 537 is a composite number.
  • Prime factorization: 537 = 3 x 179
  • 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 537 has exactly 4 factors.
  • Factors of 537: 1, 3, 179, 537
  • Factor pairs: 537 = 1 x 537 or 3 x 179
  • 537 has no square factors that allow its square root to be simplified. √537 ≈ 23.17326

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537 Factors

536 Family Reunion

Last week I attended a family reunion. My uncle Bob showed me a very clever way that helps him remember the number of children that my dad and each of his siblings had.

How Many Children

In case you are wondering, I was one of Leonard’s fifteen kids: He and his first wife had 4 children. They divorced. He met my mom who already had a child of her own. They married and had 6 children. She died. Then after he married my step-mother who already had two grown children, they had two more.

  • 536 is a composite number.
  • Prime factorization: 536 = 2 x 2 x 2 x 67, which can be written 536 = (2^3) x 67
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 536 has exactly 8 factors.
  • Factors of 536: 1, 2, 4, 8, 67, 134, 268, 536
  • Factor pairs: 536 = 1 x 536, 2 x 268, 4 x 134, or 8 x 67
  • Taking the factor pair with the largest square number factor, we get √536 = (√4)(√134) = 2√134 ≈ 23.15167

535 and Level 6

535 is the hypotenuse of the Pythagorean triple 321-428-535. Can you find the greatest common factor of those three numbers?

535 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 535 is a composite number.
  • Prime factorization: 535 = 5 x 107
  • 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 535 has exactly 4 factors.
  • Factors of 535: 1, 5, 107, 535
  • Factor pairs: 535 = 1 x 535 or 5 x 107
  • 535 has no square factors that allow its square root to be simplified. √535 ≈ 23.130067

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535 Logic

534 and Level 5

534 is made of three consecutive digits so it can be evenly divided by three.

534 is the sum of consecutive primes: 127 + 131 + 137 + 139 = 534.

534 is the hypotenuse of the Pythagorean triple 234-480-534. Can you find the greatest common factor of those three numbers?

534 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 534 is a composite number.
  • Prime factorization: 534 = 2 x 3 x 89
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 534 has exactly 8 factors.
  • Factors of 534: 1, 2, 3, 6, 89, 178, 267, 534
  • Factor pairs: 534 = 1 x 534, 2 x 267, 3 x 178, or 6 x 89
  • 534 has no square factors that allow its square root to be simplified. √534 ≈ 23.108440.

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534 Logic

533 and Level 4

533 is the sum of consecutive primes two different ways: 173 + 179 + 181 = 533 = 101 + 103 + 107 + 109 + 113.

533 = (23^2) + (2^2), and 533 = (22^2) + (7^2)

533 is the hypotenuse of four Pythagorean triples. Some of the triples have a greatest common factor greater than one, and the rest are primitive. Which are which?

  • 92-525-533
  • 117-520-533
  • 205-492-533
  • 308-435-533

533 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 533 is a composite number.
  • Prime factorization: 533 = 13 x 41
  • 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 533 has exactly 4 factors.
  • Factors of 533: 1, 13, 41, 533
  • Factor pairs: 533 = 1 x 533 or 13 x 41
  • 533 has no square factors that allow its square root to be simplified. √533 ≈ 23.08679

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533 Logic

532 and Level 3

532 is the sum of consecutive prime numbers 263 and 269.

532 is also the 19th pentagonal number.

532 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 532 is a composite number.
  • Prime factorization: 532 = 2 x 2 x 7 x 19, which can be written 532 = (2^2) x 7 x 19
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 532 has exactly 12 factors.
  • Factors of 532: 1, 2, 4, 7, 14, 19, 28, 38, 76, 133, 266, 532
  • Factor pairs: 532 = 1 x 532, 2 x 266, 4 x 133, 7 x 76, 14 x 38, or 19 x 28
  • Taking the factor pair with the largest square number factor, we get √532 = (√4)(√133) = 2√133 ≈ 23.065125

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

532 Factors

 

531 and Level 2

5 + 3 + 1 = 9 so 531 is divisible by 3 and by 9. Because it can be evenly divided by 9, it’s square root can be reduced.

531 ÷ 9 = 59, a prime number which obviously has no square factors. Thus √531 = (√9)(√59) = 3√59 in its most reduced form.

531 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 531 is a composite number.
  • Prime factorization: 531 = 3 x 3 x 59, which can be written 531 = (3^2) x 59
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 531 has exactly 6 factors.
  • Factors of 531: 1, 3, 9, 59, 177, 531
  • Factor pairs: 531 = 1 x 531, 3 x 177, or 9 x 59
  • Taking the factor pair with the largest square number factor, we get √531 = (√9)(√59) = 3√59 ≈ 23.04343724

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531 Factors

 

530 and Level 1

The first three perfect numbers are 6, 28, and 496. Their sum is 530.

Can you find the greatest common factors for each of these Pythagorean triples?

  • 46-528-530
  • 192-494-530
  • 280-450-530
  • 318-424-530

530 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 530 is a composite number.
  • Prime factorization: 530 = 2 x 5 x 53
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 530 has exactly 8 factors.
  • Factors of 530: 1, 2, 5, 10, 53, 106, 265, 530
  • Factor pairs: 530 = 1 x 530, 2 x 265, 5 x 106, or 10 x 53
  • 530 has no square factors that allow its square root to be simplified. √530 ≈ 23.02172886644

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530 Factors

Is 529 an Interesting Number?

  • 529 is a composite number.
  • Prime factorization: 529 = 23^2
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 529 has exactly 3 factors.
  • Factors of 529: 1, 23, 529
  • Factor pairs: 529 = 1 x 529 or 23 x 23
  • 529 is a perfect square. √529 = 23

In the United States you can save money for college in a tax-free 529 College Savings Plan. If you use a 529 Plan, when you are 23, you could be making a good, square income. That is how I remember that 529 = 23².

529 dots can be made into a square, but is 529 an interesting number?

23 is the 9th prime number, and 529 is the 9th number with exactly 3 factors. Is that interesting?

What about the fact that 529 is palindrome 121 in BASE 22 because 1(22²) + 2(22) + 1(1) = 529?

Or that 529 can be written as the sum of three squares three ways and each of those ways contain exactly TWO elements of the set {18², , } plus one other square number?

  • 22² + + = 529
  • 18² + 14² + = 529
  • 18² + 13² + = 529

Wikipedia informs us that 529 is the 12th centered octagonal number because  4(12²) – 4(12) + 1 = 529.

Do any of those reasons make 529 an interesting number? Is ANY number interesting?

Recently when I wrote that 526 is a centered pentagonal number, Steve Morris of Blog, Blogger, Bloggest commented:

It’s intriguing how so many numbers have interesting or special properties. I used to think that there was something magical about this, that these patterns were somehow telling us something deep about the universe. Like, why is 3.14159… the value it is?

Nowadays I think that these special properties are things we invented. For instance, a centred pentagonal number is interesting if you think it is, and isn’t if you don’t think so. Some numbers have particularly important or curious properties (prime numbers for instance), but they are only interesting because we think they are!

What do you think?

I think that much of mathematics was discovered by astronomers, physicists, and even philosophers, and the patterns in mathematics do tell us something deep about the universe, but all of those mathematical properties existed before they were discovered. They are eternal principles that we each understand to the best of our abilities.

Are numbers with certain properties interesting only because some people think they are?

If the world never knew that pi is approximately equal to 3.14159…, we would still be as advanced as we are now. Pi seems like a very important number, but what if the world had never heard of pi? What if for thousands of years the world had instead used tau (τ ) which is approximately equal to 6.28318? Every important, magical discovery related to pi would still be known. Since τ = 2π, some people think we should all celebrate on June 28 by eating twice as much pie. Those people are really saying that tau is an interesting number only because pi is interesting. If we had never heard of pi, we wouldn’t eat pie to celebrate either day. As it is, most people have never heard of τ and would consider it to be a rather boring number.

Are any numbers inherently interesting?

Ancient mathematicians were fascinated that a series of dots could be made into different shapes. Sometimes a certain number of dots could be made into a triangle, or a square, or even a pentagon. A string of dots that couldn’t be made into a 2-dimensional rectangle represented a prime number. The fact that the ancients were interested in the shape of numbers makes them even more interesting to me.

It doesn’t matter what language you use or even what number base you use, 529 ♦’s can be arranged into a perfectly formed square just as ♦ ♦ ♦ ♦ can be. Most people can understand square numbers or even cubed numbers. However, for some people triangular numbers and pentagonal numbers are just mind-boggling, and they won’t consider such numbers to be interesting at all.

I don’t know if the ancients were aware of CENTERED pentagonal numbers, but I am fascinated by the fact that this is a shape that CAN consist of a prime number of dots (31, 181, 331, for example.)

My husband often says that nothing is boring. Two different people can listen to the same talk. One of the listeners might be moved to tears or inspired to action while the other person is bored out of his mind. It isn’t the subject matter that decides how people will react, it’s the people themselves.

Mathematician G. H. Hardy is an inspiration to me. He said, “Nothing I have ever done is of the slightest practical use.” He studied mathematical topics that interested him without regard to their usefulness. How could he potentially waste Ramanujan’s great mind on something as useless as partition theory. How did he convince Cambridge University to pay him while he obsessively explored this unimportant topic especially when he boasted that it didn’t have the slightest practical use? I don’t know how he did that, but it turns out that partition theory was not simply recreational mathematics; it does have a practical purpose! From G. H. Hardy I have learned that it is okay to explore topics that interest me even if they NEVER have a practical purpose. I haven’t figured out how to get someone to pay me to explore any unimportant topics, but I can still explore them to my heart’s content:

I often notice if the square root of a number can be reduced or not. I did not invent the idea of reducing square roots, but I’ve decided it is an interesting topic. I don’t know if anybody else looks for consecutive reducible square roots or calculates what percentage of numbers have reducible square roots, but I find it interesting, and I explore it. If you google reducible square roots, chances are that much of what you see will have been created by me.

Some mathematicians spend time trying to prove the Riemann hypothesis, but that topic doesn’t interest me as much as other topics because that is what I’ve decided.

What do you think? What makes a number interesting or not? What makes a mathematical topic interesting or not?