525 and Level 3

525 = (23 + 2)(23 – 2) = (23^2) – (2^2) makes 525 the longer leg in what primitive Pythagorean triple?

Why do people enjoy number puzzles? A mathemagician friend once stated that a key reason was loving to solve mysteries.

I find making and solving puzzles to be quite relaxing. Solve the mystery or relax a little finding the factors of this puzzle:

525 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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  • 525 is a composite number.
  • Prime factorization: 525 = 3 x 5 x 5 x 7, which can be written 525 = 2 x (5^2) x 7
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 525 has exactly 12 factors.
  • Factors of 525: 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525
  • Factor pairs: 525 = 1 x 525, 3 x 175, 5 x 105, 7 x 75, 15 x 35, or 21 x 25
  • Taking the factor pair with the largest square number factor, we get √525 = (√25)(√21) = 5√21 ≈ 22.912878

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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.

525 Factors

524 and Level 2

24 is divisible by 4 and that means 524 is also divisible by 4.

Numbers that are divisible by 4 can have their square roots reduced. 524 ÷ 4 = 131, a prime number whose only square factor is 1, so √524 = (√4)(√131) = 2√131.

524 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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  • 524 is a composite number.
  • Prime factorization: 524 = 2 x 2 x 131, which can be written 524 = (2^2) x 131
  • 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 524 has exactly 6 factors.
  • Factors of 524: 1, 2, 4, 131, 262, 524
  • Factor pairs: 524 = 1 x 524, 2 x 262, or 4 x 131
  • Taking the factor pair with the largest square number factor, we get √524 = (√4)(√131) = 2√131 ≈ 22.891046

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524 Factors

523 and Level 1

523 = 61 + 67 + 71 + 73 + 79 + 83 + 89 which are all the prime numbers between 60 and 96.

521 and 523 are twin primes.

523 is the 99th prime number. It will take longer than ever before for another prime number to be featured on this blog.

The prime gap is defined as the difference between two consecutive prime numbers. The gap between this prime number and the next one is greater than between any two previous prime numbers.

GAP 100 Primes

Here is today’s puzzle:

523 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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  • 523 is a prime number.
  • Prime factorization: 523 is prime.
  • The exponent of prime number 523 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 523 has exactly 2 factors.
  • Factors of 523: 1, 523
  • Factor pairs: 523 = 1 x 523
  • 523 has no square factors that allow its square root to be simplified. √523 ≈ 22.869193

How do we know that 523 is a prime number? If 523 were not a prime number, then it would be divisible by at least one prime number less than or equal to √523 ≈ 22.9. Since 523 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 523 is a prime number.

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523 Factors

522 Gustáv Forgon and Mária Csörnök

I’ll write about the family of Gustáv Forgon and Mária Csörnök after I write a little bit about the number 522.

522 = 73 + 79 + 83 + 89 + 97 + 101 which is all the prime numbers between 72 and 102.

522 is the hypotenuse of the Pythagorean triple 360-378-522.

  • 522 is a composite number.
  • Prime factorization: 522 = 2 x 3 x 3 x 29, which can be written 522 = 2 x (3^2) x 29
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 522 has exactly 12 factors.
  • Factors of 516: 1, 2, 3, 6, 9, 18, 29, 58, 87, 174, 261, 522
  • Factor pairs: 522 = 1 x 522, 2 x 261, 3 x 174, 6 x 87, 9 x 58, or 18 x 29
  • Taking the factor pair with the largest square number factor, we get √522 = (√9)(√58) = 3√58 ≈ 22.8473193

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Gustáv Forgon was two years younger than my husband’s second great-grandmother, Erzsébet Forgon. They were only seventh cousins, but most likely they still knew each other quite well as they both had the same surname and grew up as part of one of the most prominent noble families in the little Hungarian village called Mihályfalva.

When Gustáv grew up, he married. His marriage record is the third record on the page below and states that his marriage occurred in 1873 on February 12. The record states that the groom was the noble Gusztáv Forgon, the son of the late noble Miklós Forgon and the noble Sarlotta Bodon. The groom was born and raised in Mihályfalva and was 25 years old. The bride was Mária Csörnök, daughter of Márton Csörnök and Zsuzsánna Miko. She was born and raised in Alsó-Vály and was 17 years old on their wedding day. Click on the record to see it better.15

The couple settled in  Alsó-Vály where they had TWELVE children born before 1896.

1st. Their first son, Ignácz Gusztáv Forgon, was born on 10 February 1875 and baptized two days later. His birth is the 5th entry on the page below. They lived in house #3 in Alsó-Vály.

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2nd. Curiously they named their second son Gusztáv when he was born on 25 August 1876 and baptized two days later. His birth is the 3rd entry on the page below.

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3rd. On 5 March 1879 the couple was blessed to have a daughter. They named her Apollónia Forgon, which was the same name as her godmother. Apollónia was christened two days after she was born as indicated on the 6th entry of the year. There is also a comment in the right margin: +1922 is all that I can read of it. It most likely indicates that she lived until 1922.

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On 10 April 1881 Mária’s father, Márton Csörnök, died. He had been very weak for a while. Her parents had been married for 42 of his 62 3/4 years.

4th & 5th. On 19 May 1881 Gustáv Forgon and Mária Csörnök had twin boys! They named them István and Pál. The boys were christened the same day they were born as recorded on entries 7 and 8 below.

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Sadly István was very weak and died four days later on 1881 May 24. His death record is number 17, very close to the middle of the page.

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6th. Gizella was born on 11 March 1884 and baptized the next day. Her christening is the next to the last entry below.

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7th and 8th. Gustáv Forgon and Mária Csörnök had another set of twins born on 13 April 1886. This time the twins were a boy and a girl, István and Mária. Their births are the 9th and 10th entry. Their deaths also came too early and are listed in the margins.

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This István was also very weak and died when he was only 10 days old on the 27 April 1886. His death record is third from the bottom of the page.

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Maria lived a little more than 9 months more than her twin, István, did. She died 1887 January 25 and was buried the next day. Her cause of death was listed as sínlődés. Online dictionaries were no help translating this word, but my very old and priceless Hungarian-English dictionary that a genealogist friend gave me equates the verb sínlődni and sínleni which means to be sickly, to be broken down in health, to languish. The record of her death is second from the top of the page.

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9th. A daughter, Irma, was born on 23 January 1888 and baptized the next day. She was the third baby christened in 1888.

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On 2 March 1890 Mária’s mother, Zsuzsánna Miko, died. Her death record stated that her mother was 69 years, 11 months, and 13 days old when she died. That was very important information because I could not find Márton Csörnök and Zsuzsánna Miko marriage record to learn the names of Zsuzsánna’s parents, and there were several people named Zsuzsánna Miko in the area. Now I know exactly who she is!

10th. The family’s house number changed from #3 to #4 when László was born 28 June 1890. His baptism was on 3 July as indicated in the next to last entry on the page below. I know for sure that László grew up, married, and now has many descendants.

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11th. The family’s house number is now #5. The family welcomed another little boy that they named István. He was born on 17 March 1894 and was baptized three days later as recorded on the 5th entry below. His death later that year is indicated in the margin as well.

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István died 17 October 1894 and was buried two days later. This István Forgon, age 5 months, died from weakness and was only the 15th death in the area that year.

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The record that was 3rd from the last on the same page (the 1st death record in 1895) is the death record for Gustáv’s widowed mother, Bodon Sarlolta, as it is spelled on this record. She was 72 years old when she died on 15 January 1895, and was buried two days later.

12th. Still living in house #5, the family welcomed Lajos who was born on 30 September 1895 and christened the next day. His was the 21st birth recorded in the book that year.

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To summarize I’ve made a chart showing the children born to Gustáv Forgon and Mária Csörnök from 1875 to 1895:

I enjoy using old records to piece together a family to understand some of what they went through together. Imagining their joy when they married or had a newborn baby as well as their struggles and trials when a loved one died makes them become more than just a name and a date to me. I hope you enjoyed reading about this noble Hungarian family.

521 and Level 6

521 = 20² + 11², and 521  is the hypotenuse of the primitive Pythagorean triple 279-440-521.

521 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 521 is a prime number.
  • Prime factorization: 521 is prime.
  • The exponent of prime number 521 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 521 has exactly 2 factors.
  • Factors of 521: 1, 521
  • Factor pairs: 521 = 1 x 521
  • 521 has no square factors that allow its square root to be simplified. √521 ≈ 22.8254244

How do we know that 521 is a prime number? If 521 were not a prime number, then it would be divisible by at least one prime number less than or equal to √521 ≈ 22.8. Since 521 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 521 is a prime number.

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521 Logic

520 and Level 5

520 is the hypotenuse of four Pythagorean triples. Can you find the greatest common factors of each of these triples:

  • 128-504-520
  • 200-480-520
  • 264-448-520
  • 312-416-520

520 = (23^2) – (3^2).

520 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 520 is a composite number.
  • Prime factorization: 520 = 2 x 2 x 2 x 5 x 13, which can be written 520 = (2^3) x 5 x 13
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 520 has exactly 16 factors.
  • Factors of 520: 1, 2, 4, 5, 8, 10, 13, 20, 26, 40, 52, 65, 104, 130, 260, 520
  • Factor pairs: 520 = 1 x 520, 2 x 260, 4 x 130, 5 x 104, 8 x 65, 10 x 52, 13 x 40, or 20 x 26
  • Taking the factor pair with the largest square number factor, we get √520 = (√4)(√130) = 2√130 ≈ 22.8035085

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519 and Level 4

519 = 167 + 173 + 179, three consecutive primes. The difference between each of those prime numbers is 6 so one of them is also one of the factors of 519. The other factor that pairs up with it should be obvious, too.

519 is the hypotenuse of the Pythagorean triple 156-495-519. Can you find the greatest common factor of those three numbers?

519 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 519 is a composite number.
  • Prime factorization: 519 = 3 x 173
  • 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 519 has exactly 4 factors.
  • Factors of 519: 1, 3, 173, 519
  • Factor pairs: 519 = 1 x 519 or 3 x 173
  • 519 has no square factors that allow its square root to be simplified. √519 ≈ 22.781571

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519 Logic

518 and Level 3

Finding something interesting about the number 518 was as easy as 1-2-3.

518 = (5^1) + (1^2) + (8^3). Thank you OEIS.org for that fun fact.

518 is also the hypotenuse of the Pythagorean triple 168-490-518. Can you find the greatest common factors of those three numbers?

518 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 518 is a composite number.
  • Prime factorization: 518 = 2 x 7 x 37
  • 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 518 has exactly 8 factors.
  • Factors of 518: 1, 2, 7, 14, 37, 74, 259, 518
  • Factor pairs: 518 = 1 x 518, 2 x 259, 7 x 74, or 14 x 37
  • 518 has no square factors that allow its square root to be simplified. √518 ≈ 22.759613

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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.

518 Factors

517 and Level 2

5 – 1 + 7 = 11 so 517 can be evenly divided by 11.

517 = 97 + 101 + 103 + 107 + 109 which is all the prime numbers between 90 and 112.

517 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 517 is a composite number.
  • Prime factorization: 517 = 11 x 47
  • 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 517 has exactly 4 factors.
  • Factors of 517: 1, 11, 47, 517
  • Factor pairs: 517 = 1 x 517 or 11 x 47
  • 517 has no square factors that allow its square root to be simplified. √517 ≈ 22.737634

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517 Factors

516 Is an Untouchable Number

Look at this chart of the sum of the factors for the numbers 1 – 25:

sum of factors

If we made the chart infinitely long using every counting number as n, there are certain numbers like 2, 5, 52, 88, and 96 that will NEVER appear in either column C or column D. Those numbers are called untouchable numbers, and 516 is one of them. Even though there are relatively few untouchable numbers, Paul Erdős proved that there are infinitely many of them.

516 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 516 is a composite number.
  • Prime factorization: 516 = 2 x 2 x 3 x 43, which can be written 516 = (2^2) x 3 x 43
  • 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 516 has exactly 12 factors.
  • Factors of 516: 1, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516
  • Factor pairs: 516 = 1 x 516, 2 x 258, 3 x 172, 4 x 129, 6 x 86, or 12 x 43
  • Taking the factor pair with the largest square number factor, we get √516 = (√4)(√129) = 2√129 ≈ 22.715633382

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516 Factors