Perfect Number 496 and Level 2

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  • 6 is the first perfect number because 1 + 2 + 3 = 6. Also notice that (2^1)(2^2 – 1) = 2 x 3 = 6.
  • 28 is the second perfect number because 1 + 2 + 4 + 7 + 14 = 28. Hmm… (2^2)(2^3 – 1) = 4 x 7 = 28.
  • (2^3)(2^4 – 1) = 8 x 15 = 120 is NOT a perfect number because 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 = 240 which is two times what it needs to be.
  • 496 is (2^4)(2^5 – 1) = 16 x 31. So why is 496 the third perfect number? Everything you need to know to figure out the answer to that question can be found somewhere in this post.
  • 2016 is (2^5)(2^6 – 1) = 32 x 63, and 2016 is also NOT a perfect number.

6, 28, and 496 are all triangular numbers as well as hexagonal numbers, but 120 and 2016 can also make that claim.

The clues in yesterday’s Find the Factors puzzle were all perfect squares. Today’s puzzle is only a little more difficult. You can solve it, too!

496 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-18

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  • 496 is a composite number.
  • Prime factorization: 496 = 2 x 2 x 2 x 2 x 31, which can be written 496 = (2^4) x 31
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 496 has exactly 10 factors.
  • Factors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
  • Factor pairs: 496 = 1 x 496, 2 x 248, 4 x 124, 8 x 62, or 16 x 31
  • Taking the factor pair with the largest square number factor, we get √496 = (√16)(√31) = 4√31 ≈ 22.271057

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496 Factors

495 and Level 1

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495 has an almost magical property and is known as the Kaprekar constant for 3 digit numbers. What does that mean?

Take any 3 digit number that has at least 2 different digits. Write the digits from greatest to least to create a new 3-digit number. From that number subtract the same digits written in reverse order. Repeat this process, and you will get the number 495 in no more than seven iterations. This graphic shows this process applied to the number 101.

495-Kaprekar Transformation

The following puzzle can easily transform into a multiplication table if you first find all the square factors of the given clues. Go ahead give it a try!

495 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-18

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  • 495 is a composite number.
  • Prime factorization: 495 = 3 x 3 x 5 x 11, which can be written 495 = (3^2) x 5 x 11
  • 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 495 has exactly 12 factors.
  • Factors of 495: 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495
  • Factor pairs: 495 = 1 x 495, 3 x 165, 5 x 99, 9 x 55, 11 x 45, or 15 x 33
  • Taking the factor pair with the largest square number factor, we get √495 = (√9)(√55) = 3√55 ≈ 22.248595

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495 Factors

494 My First Microfilm Treasure Hunt

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494 is the hypotenuse of one Pythagorean triple: 190-456-494. What is the greatest common factor of those three numbers?

  • 494 is a composite number.
  • Prime factorization: 494 = 2 x 13 x 19
  • 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 494 has exactly 8 factors.
  • Factors of 494: 1, 2, 13, 19, 26, 38, 247, 494
  • Factor pairs: 494 = 1 x 494, 2 x 247, 13 x 38, or 19 x 26
  • 494 has no square factors that allow its square root to be simplified. √494 ≈ 22.22611

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Years ago my husband and I wanted to research his family tree so I decided to visit the Family History library in downtown Salt Lake City on 8 April 2010. At that time we knew the names of his four grandparents, his eight great-grandparents, and one great-great-grandfather. We knew all of these people were born in Hungary. The information we had was that two of his grandparents were born in a little town called Gyoma in what is now eastern Hungary. We had no idea where his other two grandparents were born. This day was the first time I ever looked at any Hungarian genealogical records. I knew maybe ten words in Hungarian, and I had never even seen similar records in English.

The records were recorded on microfilm which were sorted into Roman Catholic, Reformed Hungarian, Lutheran, and Jewish records. I had no idea what religion his ancestors were, but based on the number of microfilms available for each religion in Gyoma, chances were that they belonged to the Reformed Church. I found a microfilm with Kereszteltek (christening) records from 1883-1895. A volunteer showed me how to put the microfilm on a the reader, and I started looking. I made notes of which records interested me. It was so exciting to find records that had names of people I had heard stories about. It was my intention to photocopy as many family records as I could, scan them into my home computer, and email them to my son, David, who speaks Hungarian fluently but lived in Qatar at the time. However, when I went to make copies, I was pleasantly surprised to learn that I could actually copy the records directly onto a flash drive!

I emailed my son that the five hours I spent at the family history library were well spent. I didn’t find any of the christening records I was expecting to find but found about thirteen records of his ancestor’s siblings. I attached the records to the email and waited for morning to arrive in Qatar so he could reply.

The next day he emailed me back, “I only had time to look at the first four (records). I’ll check the rest later. I’ve written some notes below, but I should let you know that you basically just found four people who aren’t related to us.” He then wrote in English what each of the records said.

Later he emailed me, “To continue the bad news, Now that I look at all of them, I can see that they (the great-grandparents) are all listed as godparents. This should explain why you didn’t find much of what you were actually looking for. Now you know, and should be able to look for names in the right column.”

He sent me translations of the page headings so I wouldn’t go wrong in the future. The christening records were two pages wide. Here are the headings with his translations for the first page:

And here are the headings with translations for the 2nd page.

So there you have it. Since I knew so little Hungarian and so little about how christening records are organized, I thought the godparents were the parents.

I had to wait a whole week before I could go back to the library, but this first visit was not a total bust. We still learned a few things about my husband’s great-grandfather, Dániel Finta, that we didn’t know before. We learned that he worked in a factory making shoes because his profession was given next to his name on at least one record. We learned that Dániel was asked on several occasions to be a godfather. Sometimes his wife was the godmother with him, and sometimes his mother was. Because I found these records we now knew his mother’s name, Sára Bíró, as well. We also learned that Dániel belonged to the Reformed Church and his wife, Emília Pribelszky, was Lutheran.

I was grateful for what we had learned and anxious to return again.

How successful were you the first time you looked into your family history? If you were discouraged, please give it another try. It is so worth it. If you were successful, you know exactly what I mean.

493 and Level 6

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493 is the sum of some consecutive prime numbers. Check the comments to see who posts those consecutive primes first, one of my readers or me.

David Radcliffe posted this 493-related gif on twitter:

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493 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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  • 493 is a composite number.
  • Prime factorization: 493 = 17 x 29
  • 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 493 has exactly 4 factors.
  • Factors of 493: 1, 17, 29, 493
  • Factor pairs: 493 = 1 x 493 or 17 x 29
  • 493 has no square factors that allow its square root to be simplified. √493 ≈ 22.203603

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493 Logic

Simplifying √492 and Level 5

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492 is the sum of some consecutive prime numbers (actually two different ways: see the comments). I’ll list those primes in the comments in about a week unless somebody else beats me to it. (abyssbrain beat me to it.)

Simplifying the square root of 492 is as easy as 1-2-3:

92 can be evenly divided by 4, so 492 is also divisible by 4. If I wanted to find √492, I would make a little cake and divide 492 by 4.

492 cake

123 is not divisible by 4 or by 9, but it is divisible by 3 so I will do that division next and get a quotient of 41, a prime number. Now I will take the square root of everything on the outside of the cake and multiply them together: √492 = (√4)(√3)√(41) = (√4)(√123) =2√123

492 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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  • 492 is a composite number.
  • Prime factorization: 492 = 2 x 2 x 3 x 41, which can be written 492 = (2^2) x 3 x 41
  • 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 492 has exactly 12 factors.
  • Factors of 492: 1, 2, 3, 4, 6, 12, 41, 82, 123, 164, 246, 492
  • Factor pairs: 492 = 1 x 492, 2 x 246, 3 x 164, 4 x 123, 6 x 82, or 12 x 41
  • Taking the factor pair with the largest square number factor, we get √492 = (√4)(√123) = 2√123 ≈ 22.181073

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492 Logic

491 and Level 4

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491 is the sum of consecutive primes two different ways. Check the comments to see if any of my readers list what those consecutive primes are.

491 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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  • 491 is a prime number.
  • Prime factorization: 491 is prime.
  • The exponent of prime number 491 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 491 has exactly 2 factors.
  • Factors of 491: 1, 491
  • Factor pairs: 491 = 1 x 491
  • 491 has no square factors that allow its square root to be simplified. √491 ≈ 22.1585198

How do we know that 491 is a prime number? If 491 were not a prime number, then it would be divisible by at least one prime number less than or equal to √491 ≈ 22.159. Since 491 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 491 is a prime number.

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491 Logic

490 and Level 3

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490 is seventy times seven. A famous Bible verse mentioning this number is Matthew 18: 21 – 22.

This is the King James Version: “Then came Peter to him, and said, Lord, how oft shall my brother sin against me, and I forgive him? till seven times? Jesus saith unto him, I say not unto thee, Until seven times: but, Until seventy times seven.”

Most other Bible translation also have seventy times seven, but a few translations have only seventy seven. You can compare several different translations of Matthew 18: 22 here. Regardless of the translation, forgiveness is much more important than keeping count especially since all of us need to be forgiven as well.

To find √490 first notice that 490 can be evenly divided by 2, but not by 4. It also cannot be evenly divided by 9. It is divisible by 5, but since it doesn’t end in 25, 50, 75, or 00, it can only be divided by 5 once. Since it is divisible by 10, and dividing by 10 is so easy, I would make this one-layer cake:

490 cake

Then I would take the square root of everything on the outside of the cake and multiply together to get √490 = (√10)(√49) = 7√10.

490 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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  • 490 is a composite number.
  • Prime factorization: 490 = 2 x 5 x 7 x 7, which can be written 490 = 2 x 5 x (7^2)
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 490 has exactly 12 factors.
  • Factors of 490: 1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490
  • Factor pairs: 490 = 1 x 490, 2 x 245, 5 x 98, 7 x 70, 10 x 49, or 14 x 35
  • Taking the factor pair with the largest square number factor, we get √490 = (√49)(√10) = 7√10 ≈ 22.135943

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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.

490 Factors

 

 

 

489 and Level 2

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489 is the ninth octahedral number.  Its square root starts out in quite a memorable way: √489 ≈ 22.113344, but after those six decimal places, it looks as irrational as can be.

 489 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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  • 489 is a composite number.
  • Prime factorization: 489 = 3 x 163
  • 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 489 has exactly 4 factors.
  • Factors of 489: 1, 3, 163, 489
  • Factor pairs: 489 = 1 x 489 or 3 x 163
  • 489 has no square factors that allow its square root to be simplified. √489 ≈ 22.113344

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489 Factors

488 and Level 1

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I learned about a fun addition fact from OEIS.org. Look at all the 2’s in the sum and also a couple of powers of 2 in the hundred’s, ten’s, and one’s places of 488.

488

Since 488 is the sum of two squares, it is also the hypotenuse of a Pythagorean triple: 88-480-488. The greatest common factor of those three numbers should be easy to spot.

488 is also divisible by 4 so its square root can be reduced. 488 ÷ 4 = 122. We can divide 122 by 2 to get 61 to verify that 122 has no square factors, but since it doesn’t we get this one layer cake:

488 cake

We simplify the square root by taking the square root of everything on the outside of the cake: √488 = (√4)(√122) = 2√122.

488 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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  • 488 is a composite number.
  • Prime factorization: 488 = 2 x 2 x 2 x 61, which can be written 488 = (2^3) x 61
  • 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 488 has exactly 8 factors.
  • Factors of 488: 1, 2, 4, 8, 61, 122, 244, 488
  • Factor pairs: 488 = 1 x 488, 2 x 244, 4 x 122, or 8 x 61
  • Taking the factor pair with the largest square number factor, we get √488 = (√4)(√122) = 2√122 ≈ 22.09072

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488 Factors

487 The Forgon Family Tree

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Before I write about the Forgon Family, I’ll write just a little bit about the number 487 beginning with something I learned from Number Gossip:

  • 487¹ = 487, and 4 + 8 + 7 = 19.
  • 487³ = 115,501,303, and 1 + 1 + 5 + 5 + 0 + 1 + 3 + 0 + 3 = 19.

It’s pretty cool that both sums equal each other, but it’s even cooler that 487 is the smallest prime number that can make that claim.

487 = 157 + 163 + 167, so 487 is also the sum of three consecutive prime numbers.

  • 487 is a prime number.
  • Prime factorization: 487 is prime.
  • The exponent of prime number 487 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 487 has exactly 2 factors.
  • Factors of 487: 1, 487
  • Factor pairs: 487 = 1 x 487
  • 487 has no square factors that allow its square root to be simplified. √487 ≈ 22.068076

How do we know that 487 is a prime number? If 487 were not a prime number, then it would be divisible by at least one prime number less than or equal to √487 ≈ 22.068. Since 487 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 487 is a prime number.

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Forgon Andor and David

Andor Forgon and my son, David. Andor is the caretaker of the Mihály Forgon museum in Mihályfalva. David wrote, “Andor Forgon, who is a distant cousin (If I’ve done my math right we’re tenth cousins twice removed. You’d have to go back to the 1600s to find a common ancestor). Still, he had a lot of interesting information about the Forgon branch of our family and about the history of Mihályfalva.”

My husband’s second great-grandmother was named Erzsébet Forgon. She was born into Hungarian nobility in a little village called Mihályfalva in what is now southern Slovakia. Her parents were Juditha Dancs and Boldizsár Forgon.

Since Erzsébet was born into a Catholic family, we were not able to find her christening record in Mihályfalva. It was very discouraging pouring over the Reformed Church records, seeing plenty of people with the name Forgon, but not her christening record. I found the record of her conversion from Catholicism to the Reformed Church. It’s the last record on the page below. The images are small, but if you click on them, you should be able to read them much more easily.

162 Forgon Erzsébet

I was also able to find her marriage record. It is the first entry in the year 1856.

3 1856-01-24 wedding

I had almost given up hope finding her christening record. When my son and I visited Mihályfalva three years ago, he asked someone in town where a Catholic would take their children to get baptized. The town named seemed so far away. We looked online a little but did not immediately find her christening record.

Because Familysearch volunteers have indexed so many records, we were able to find Erzsébet’s 9 October 1836 christening record here. Her christening is listed near the top of the second of the two pages of the document.

I was also able to find the 5 June 1809 christening record of her father, Boldizsár son of János Forgon and Krisztina Nagy.  That baptism is the second entry in June, and his brother’s christening is listed right under his.

The 8 May 1768 christening of my husband’s 4th great grandfather, János Forgon, son of Péter Forgon and Borbála Kovács is the third entry on the first page of this document.

This 19 June 1741 document appears be the christening record of my husband’s 5th great grandfather Péter Forgon, son of István (Stephan) Forgon. It is the 7th entry on the 2nd page of the document. This christening occurred in Mihályfalva at a time when mothers were not considered important enough to list on records. Péter and his brother István who was christened 26 April 1743 (1st page; 17 entry) both converted to Catholicism.

All of these ancestors lived in Mihályfalva and the Catholic baptisms were performed in two different towns. I probably would not have found any of them if they had not been indexed and if not for the genealogical work done by one of my husband’s most important relatives. A very short account of his life follows:

One of the most famous people named Forgon was Dr. Mihály Forgon. His 22 October 1885 christening is 4th from the bottom of the first page. While he worked on his law degree he found time to compile descendant charts for the many noble families who lived in Gömör County, Hungary. After receiving his law degree, Dr. Forgon worked as a prosecutor. During World War I, he served as a reserve lieutenant on the Russian front in Poland.  About three weeks after he arrived in Poland, he was tragically and fatally shot. He was only 29 years old.

I’ve included the descendant table Mihály Forgon made for the Forgon family below. After not too many years a descendant chart becomes much too large to fit on one single sheet of paper so Mihály Forgon separated the descendant chart into three additonal tables. The earliest date on the main table is 1573, and it maps the way to the remaining tables as follows:

  • Four generations below Forgon János we have Balint who becomes the top of table #IV. (We will see my husband’s family on this table.)
  • The next generation has János who becomes the top of table #III.
  • That same generation also has Zsigmond, the father of István and Zsigmond who are at the top of table #II.

Forgon 239

Dr. Mihály Forgon name is listed near the bottom of table #I under the names of his parents, Rafáel Forgon and Erzsébet Bodon. Forgon and Bodon were both noble families and the most honored surnames in Mihályfalva.

Forgon 240

Forgon 241

My husband’s second great grandmother, Erzsébet, is listed on this fourth chart. You can see her name in the middle of the chart approaching the right hand side under Boldizsár and his wife Juditha Dancs. Erzsébet’s husband, Ferdinánd Barna, is listed just below her name.

Forgon 242 Forgon Boldizsár & Dancs Judit

One of the reasons I wanted to write about the Forgon family is because I’ve met one of its members on WordPress. The beautiful Veronika Forgon also traces her roots back to Mihályfalva to this noble family. She is the lovely model featured in these four posts:

Veronika Forgon – Hajógyári Sziget

Veronika Forgon – Buda Castle

Veronika Forgon – Margitsziget

Veronika Forgon – Kopaszi Dam

Update: When I wrote this post I wasn’t exactly sure how Veronika is related to my husband and my children, but after reading it, she contacted us, and now I know! I was thrilled to learn that she is my husband’s 11th cousin, and my children are her 11th cousins once removed.