488 and Level 1

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

—————————————————————————————————

  • 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

—————————————————————————————————

488 Factors

487 The Forgon Family Tree

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.

—————————————————

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.

Finding √486 and Level 6

Today’s Puzzle:

486 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

Here is a logical order to use the clues to find the solution to the puzzle:

486 Logic

How to Simplify the Square Root of 486:

To find √486, I like to first see if 486 can be evenly divided by square numbers 4 or 9 because about 82% of numbers that can have their square roots reduced are divisible by 4 and/or by 9. It is also very easy to tell if a number can be evenly divided by 4 or by 9.

86 is not divisible by 4, so 486 is not divisible by 4.

However, 4 + 8 + 6 = 18, a multiple of 9 so 486 is divisible by 9, and now we know for sure that its square root can be simplified. 486 ÷ 9 = 54, another multiple of 9 so we get this two-layer cake when we divide by 9 twice.

486 cake

Now take the square root of everything on the outside of the cake, and we get √486 = (√9)(√9)(√6) = 9√6.

Factors of 486:

  • 486 is a composite number.
  • Prime factorization: 486 = 2 x 3 x 3 x 3 x 3 x 3, which can be written 486 = 2 x 3⁵.
  • The exponents in the prime factorization are 1 and 5. Adding one to each and multiplying we get (1 + 1)(5 + 1) = 2 x 6 = 12. Therefore 486 has exactly 12 factors.
  • Factors of 486: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486
  • Factor pairs: 486 = 1 x 486, 2 x 243, 3 x 162, 6 x 81, 9 x 54, or 18 x 27
  • Taking the factor pair with the largest square number factor, we get √486 = (√81)(√6) = 9√6 ≈ 22.04540

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

486 has eight factor pairs. One of the factor pairs adds up to 45, and a different one subtracts to 45. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

485 and Level 5

485 is the hypotenuse of four Pythagorean triples. Which ones are primitive and which ones aren’t?

  • 44-483-485
  • 93-476-485
  • 291-388-485
  • 325-360-485

485 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

—————————————————————————————————

  • 485 is a composite number.
  • Prime factorization: 485 = 5 x 97
  • 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 485 has exactly 4 factors.
  • Factors of 485: 1, 5, 97, 485
  • Factor pairs: 485 = 1 x 485 or 5 x 97
  • 485 has no square factors that allow its square root to be simplified. √485 ≈ 22.0227155

—————————————————————————————————

485 Logic

484 and Level 4

484 is the sum of 14 consecutive primes, and the first of those primes is one of its factors!

11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 = 484

484 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

—————————————————————————————————

  • 484 is a composite number.
  • Prime factorization: 484 = 2 x 2 x 11 x 11, which can be written 484 = (2^2) x (11^2)
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 x 3 = 9. Therefore 484 has exactly 9 factors.
  • Factors of 484: 1, 2, 4, 11, 22, 44, 121, 242, 484
  • Factor pairs: 484 = 1 x 484, 2 x 242, 4 x 121, 11 x 44, or 22 x 22
  • 484 is a perfect square. √484 = 22

—————————————————————————————————

484 Logic

483 and Level 3

4 + 8 = 12, a multiple of 3, so 483 can be evenly divided by 3. (It wasn’t necessary to add 4 + 8 + 3 because we already know 3 is divisible by 3.)

We can easily see that 483 is divisible by 7 if we apply the divisibility trick for 7. Separate the last number, 3, from the rest, double it, and subtract the double from the remaining numbers: 48 – (3 x 2) = 48 – 6 = 42, which is a multiple of 7. Thus 483 is divisible by 7.

Since 483 is 21 x 23, we can predict that 484 will be 22 x 22.

483 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

—————————————————————————————————

  • 483 is a composite number.
  • Prime factorization: 483 = 3 x 7 x 23
  • 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 483 has exactly 8 factors.
  • Factors of 483: 1, 3, 7, 21, 23, 69, 161, 483
  • Factor pairs: 483 = 1 x 483, 3 x 161, 7 x 69, or 21 x 23
  • 483 has no square factors that allow its square root to be simplified. √483 ≈ 21.97726

—————————————————————————————————

483 Factors

482 and Level 2

Can you determine the greatest common factor of the three numbers in this Pythagorean triple: 240-418-482? Hint: it is one of the factors of 482 listed below the puzzle.

482 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

—————————————————————————————————

  • 482 is a composite number.
  • Prime factorization: 482 = 2 x 241
  • 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 482 has exactly 4 factors.
  • Factors of 482: 1, 2, 241, 482
  • Factor pairs: 482 = 1 x 482 or 2 x 241
  • 482 has no square factors that allow its square root to be simplified. √482 ≈ 21.954498

—————————————————————————————————

 482 Factors

481 and Level 1

Can you find the greatest common factor of the three numbers in this Pythagorean triple 156-455-481? Or in Pythagorean triple 185-444-481?

481 is also the hypotenuse in two primitive Pythagorean triples: 319-360-481 and 31-480-481. Since they are primitive triples, their greatest common factor is 1.

481 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

—————————————————————————————————

  • 481 is a composite number.
  • Prime factorization: 481 = 13 x 37
  • 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 481 has exactly 4 factors.
  • Factors of 481: 1, 13, 37, 481
  • Factor pairs: 481 = 1 x 481 or 13 x 37
  • 481 has no square factors that allow its square root to be simplified. √481 ≈ 21.93171

—————————————————————————————————

481 Factors

480 The Very Inspiring Blogger Award

I Was Nominated for 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.

Who I Nominate for the Very Inspiring Blogger 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.

A Factor Tree for 480:

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

Factors of 480:

  • 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

Sum Difference Puzzles:

30 has four factor pairs. One of those pairs adds up to 13, and  another one subtracts to 13. Put the factors in the appropriate boxes in the first puzzle.

480 has twelve factor pairs. One of the factor pairs adds up to 52, and a different one subtracts to 52. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 480:

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.

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 that are divisible by 4, it is in 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!

 

479 and Level 6

479 is the sum of nine consecutive prime numbers.

37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 = 479.

479 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-04-27

—————————————————————————————————

  • 479 is a prime number.
  • Prime factorization: 479 is prime.
  • The exponent of prime number 479 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 479 has exactly 2 factors.
  • Factors of 479: 1, 479
  • Factor pairs: 479 = 1 x 479
  • 479 has no square factors that allow its square root to be simplified. √479 ≈ 21.88607

How do we know that 479 is a prime number? If 479 were not a prime number, then it would be divisible by at least one prime number less than or equal to √479 ≈ 21.8861. Since 479 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 479 is a prime number.

—————————————————————————————————

479 Logic