474 and Level 1

474 is the sum of some consecutive prime numbers which one of my readers listed in the comments. Try to figure out what they are yourself before checking there to see if you were right.

474 Puzzle

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

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  • 474 is a composite number.
  • Prime factorization: 474 = 2 x 3 x 79
  • 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 474 has exactly 8 factors.
  • Factors of 474: 1, 2, 3, 6, 79, 158, 237, 474
  • Factor pairs: 474 = 1 x 474, 2 x 237, 3 x 158, or 6 x 79
  • 474 has no square factors that allow its square root to be simplified. √474 ≈ 21.77154

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474 Factors

Just a Little 473 Cake

Today is our wedding anniversary. Today I tried to spend my time with my husband instead of the computer. Hence this is a very short and plain post.

The middle digit of 473 equals the sum of the other two digits which means that 473 can be evenly divided by 11.

Five years from now I could make one of these cakes for our anniversary!

473 is the sum of some consecutive prime numbers two different ways. See if you can find them yourself. Then check the comments to see if you were right.

  • 473 is a composite number.
  • Prime factorization: 473 = 11 x 43
  • 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 473 has exactly 4 factors.
  • Factors of 473: 1, 11, 43, 473
  • Factor pairs: 473 = 1 x 473 or 11 x 43
  • 473 has no square factors that allow its square root to be simplified. √473 ≈ 21.74856

How to Simplify √472

472 is divisible by 4 because 72 can be evenly divided by 4, and that means that √472 can be simplified. This is the method I use to reduce it:

First I divide 472 by 4 and get 118. If I recognize that 118 has no square factors, I simply can say √472 = (√4)(√118) = 2√118.

If I don’t recognize that 118 has no other square factors, I can divide it by 2 to get 59 which is a prime number and definitely has no square factors besides the number 1.

472 cake

Then I take the square root of everything on the outside of the cake to get √472 = (√4)√(2 x 59) = (√4)(√118) = 2√118

Here is today’s puzzle:

472 Puzzle

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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  • 472 is a composite number.
  • Prime factorization: 472 = 2 x 2 x 2 x 59, which can be written 472 = (2^3) x 59
  • 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 472 has exactly 8 factors.
  • Factors of 472: 1, 2, 4, 8, 59, 118, 236, 472
  • Factor pairs: 472 = 1 x 472, 2 x 236, 4 x 118, or 8 x 59
  • Taking the factor pair with the largest square number factor, we get √472 = (√4)(√118) = 2√118 ≈ 21.72556

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472 Logic

471 and Level 5

471 is the sum of some consecutive prime numbers. One of my readers has listed those primes in the comments.

471 Puzzle

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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  • 471 is a composite number.
  • Prime factorization: 471 = 3 x 157
  • 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 471 has exactly 4 factors.
  • Factors of 471: 1, 3, 157, 471
  • Factor pairs: 471 = 1 x 471 or 3 x 471
  • 471 has no square factors that allow its square root to be simplified. √471 ≈ 21.70253

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471 Logic

470 Greatest Common Factors of Pythagorean Triples.

  • 470 is a composite number.
  • Prime factorization: 470 = 2 x 5 x 47
  • 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 470 has exactly 8 factors.
  • Factors of 470: 1, 2, 5, 10, 47, 94, 235, 470
  • Factor pairs: 470 = 1 x 470, 2 x 235, 5 x 94, or 10 x 47
  • 470 has no square factors that allow its square root to be simplified. √470 ≈ 21.67948

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470 is the hypotenuse of the non-primitive Pythagorean triple 282-376-470. What is the greatest common factor of those three numbers?

The greatest common factor will always be a factor of the smallest of the three numbers, but it will also be a factor of the smallest difference between the three numbers as well. Let’s find those differences. Note: the difference between the 282 and 470 will not be the smallest difference so there is no need to find that one. We only need to find these two differences:

470 difference

In the case of this Pythagorean triple the differences are equal to each other which means that the difference, 94*, is also the greatest common factor of the three numbers! Go ahead and try dividing each number in the triple by 94. You will discover that this Pythagorean triple is just 3-4-5 multiplied by 94.

*This statement is only true of Pythagorean triples. For example the following numbers also have differences of 94, but the greatest common factor is not 94, but a factor of 94:

  1. The greatest common factor of 283-377-471 is 1.
  2. The greatest common factor of 284-378-472 is 2
  3. The greatest common factor of 329-423-517 is 47

Mathchat has written an excellent post on finding the greatest common factor of three or more numbers that can be used for all integers in general.

But as far as Pythagorean triples are concerned, anytime the corresponding differences of a Pythagorean triple are equal to each other, then that Pythagorean triple is just 3-4-5 multiplied by the difference. There are an infinite number of such triples, and 282-376-470 is just one of them.

3-4-5 Pythagorean Triple Sequence

Now remember there is an infinite number of primitive Pythagorean triples, and every one of those triples can be multiplied by each of the infinitely many counting numbers. A graphic like the one above could be made for every primitive triple followed by each of its multiples. For example 5-12-13, 10-24-26, 15- 36-39, etc. would be another infinite series of Pythagorean triples.

You could say the total number of Pythagorean triples equals infinity times infinity!

469 and Level 4

469 is the short leg in the Pythagorean triple 469-1608-1675. What is the greatest common factor of those three numbers? Hint: Don’t let the larger numbers scare you; the greatest common factor is a factor of 469, the smallest of those three numbers, and its factors are listed below the puzzle.

469 Puzzle

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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  • 469 is a composite number.
  • Prime factorization: 469 = 7 x 67
  • 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 469 has exactly 4 factors.
  • Factors of 469: 1, 7, 67, 469
  • Factor pairs: 469 = 1 x 469 or 7 x 67
  • 469 has no square factors that allow its square root to be simplified. √469 ≈ 21.6564

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469 Logic

468 and Level 3

468 is the sum of some consecutive primes. One of my readers posted what those primes are in the comments.

468 is 3333 in base 5. (Thank you OEIS.org for that cool fact.) Here’s proof going from right to left using some easy division problems. (See 3333 at the bottom.)

468 in base 5And here’s proof going from left to right using some more difficult division problems. (See 3333 at the top.)

468 from base 10 to base 5

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The last two digits of 468 can be evenly divided by 4, so 468 is also divisible by 4.

468 is made from three consecutive even numbers so it is divisible by 3. Since the middle digit of the three consecutive even numbers is divisible by 3, we know that 468 can also be evenly divided by 9.

Let’s use those two facts to simplify the square root of 468 using the modified cake method. If you’re very confident in your ability to divide, you can make a one-layer cake and simply divide 468 by 36 to get 13. Then take the square root of everything on the outside of the cake and multiply them together: √468 = (√36)(√13) = 6√13

Many people will feel more comfortable making a two layer cake by dividing first by 4 and then by 9 as illustrated below:

458 square root

Then to simplify √468, take the square root of everything on the outside of the cake and multiply those square roots together: √468 = (√4)(√9)(√13) = (2 x 3)(√13) = 6√13

You only need to know multiplication facts up to 12 x 12 to solve this factoring puzzle:

468 Puzzle

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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  • 468 is a composite number.
  • Prime factorization: 468 = 2 x 2 x 3 x 3 x 13, which can be written 468 = (2^2) x (3^2) x 13
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 468 has exactly 18 factors.
  • Factors of 468: 1, 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468
  • Factor pairs: 468 = 1 x 468, 2 x 234, 3 x 156, 4 x 117, 6 x 78, 9 x 52, 12 x 39, 13 x 36 or 18 x 26
  • Taking the factor pair with the largest square number factor, we get √468 = (√36)(√13) = 6√13 ≈ 21.6333

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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.

468 Factors

467 and Level 2

467 is part of the 22nd prime triplet, (461-463-467). The next prime triplet will not occur until (613, 617, 619).

467 Puzzle

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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  • 467 is a prime number.
  • Prime factorization: 467 is prime.
  • The exponent of prime number 467 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 467 has exactly 2 factors.
  • Factors of 467: 1, 467
  • Factor pairs: 467 = 1 x 467
  • 467 has no square factors that allow its square root to be simplified. √467 ≈ 21.61018

How do we know that 467 is a prime number? If 467 were not a prime number, then it would be divisible by at least one prime number less than or equal to √467 ≈ 21.6102. Since 467 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 467 is a prime number.

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467 Factors

466 and Level 1

466 is an interesting number whose factors are listed below today’s puzzle.

Thanks to OEIS.org, I know that 466 is 1234 in base 7. There are two different ways to change 466 from base 10 to base 7.

The first way gives the answer at the top of all the division problems and has you working from left to right. The division problems may be more difficult because division by 7 cubed and 7 squared are required, but the concept of what is happening is fairly easy to understand.

466 is 1234 in base 7 steps 1-4

For the second way, each of the division problems is quite easy to do. However working from right to left and finding the answer at the bottom of all the problems may be confusing to some people.

466 is 1234 in base 7 steps 4-1

 

Changing a number from base ten to base seven can be a bit of a challenge. However, today’s Find the Factors puzzle is as easy as they get. Every person who has learned how to multiply can find the factors for this puzzle and then make the puzzle work like a multiplication table:

466 Puzzle

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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Here is the factoring information for 466:

  • 466 is a composite number.
  • Prime factorization: 466 = 2 x 233
  • 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 466 has exactly 4 factors.
  • Factors of 466: 1, 2, 233, 466
  • Factor pairs: 466 = 1 x 466 or 2 x 233
  • 466 has no square factors that allow its square root to be simplified. √466 ≈ 21.58703

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466 Factors

 

465 Looking for Aunt Betty

 

My husband had an aunt that I had never met. In fact, he had never met her. She was the baby in her family. The rest of her family had lost contact with her 50 or 60 years ago. All they knew was that she married Herbert Bender and that the two of them had moved to Washington D. C. There may have been some unkind words spoken by them or by her, and there were some very hurt feelings. Some family members didn’t care if they ever saw or heard from her again. Nobody knew her address or phone number.

Forgetting about her just wasn’t acceptable to me so we searched for her on our very limited budget. Back in the day before the internet, when long distance phone calls were expensive, and we lived in the Hampton Roads area of Virginia, we drove up to Washington D. C.  One of the things we did when we were there was go to a phone booth and call every H. Bender in the phone book, but none of them was her husband.

Eventually, all of Betty’s brothers and sisters died except her brother, Paul. He was eight years older than she was but was the closest in age to her. Paul came to live with us in 1988, and he brought his photo albums with him. For the first time, we got to see photos of his little sister, Betty. Here are a few of those photos:

IMG_0008

The most recent picture of Betty that her brother, Paul, had.

IMG_0008_NEW

Paul and Betty working together. Betty was 5 years old. The identity of the older boy is unknown. I suggested to Paul that it was his brother, Steve, but he said it couldn’t be. “Ma never would have given him such a bad haircut.”

IMG_0007

Paul posing with his younger sister, Elizabeth (Betty).

We were so excited to see these pictures of Betty. Paul had no ill feelings toward his sister so we asked him if he would like to find her. He stated that he wanted to respect her privacy if she wanted nothing to do with the rest of the family.

Paul died in November 2005. I missed him terribly especially since, primarily, I had been the one who took care of him the last 7 1/2 years of his life. We often looked at the pictures and records he left us. There were several pictures of his folks and his siblings, his christening record from Igazfalva written in Hungarian, his passport, his naturalization record, and many other records. I eventually took public transport to downtown Salt Lake City to the Family History Library. I checked out microfilm from Gyoma, Hungary and was thrilled to find the christening record for Paul and Betty’s father, Sallai István. After several months I found the family’s genealogy all the way back to the mid 1700’s. How I wished I could have shared these records with Paul or that I could find Betty and share them with her if she were still alive.

Periodically we looked at the social security death index for Elizabeth Bender born April 7, 1921. We didn’t find her, but that was a good thing because that might mean she was still alive. One problem with knowing that for sure was that since she was a woman, her surname would be different if she ever married someone else. I loved searching through these old records and indexes. I learned that if I was in the right time and place, I could find a gold mine of records, but if I wasn’t, there was nothing to be found.

Family Search has been indexing records over the last several years. In June 2014, I was able to find this indexed marriage license record.

Elizabeth Sallai - Herbert Bender Marriage

I was tickled to find out that Herbert Bender’s occupation was a Statistician, and amused that Elizabeth Sallay said she was 22 years old and born in Cleveland, Ohio. At the time she was actually 19 years old, and she was born in Hungary.

If I had been searching through microfilm marriage records all by myself, I never would have looked in Columbus, Ohio; instead, I would have spent years searching through Cleveland marriage records. But because of an indexer, I was able to find their marriage record, and get her husband’s date of birth. That date helped me know I had found the correct person when I found his name under the social security death index and the United States Public Records. The public records gave me a phone number, but it had been disconnected. It also gave me an address. She had been born 93 years previously, and it appeared that if she was still alive, she had probably moved to a different location. I found a list of the homeowners in that Maryland neighborhood. It was obvious that the list was a little old, but I was determined to write some letters to see if anyone remembered her. I googled one of the other houses and discovered it was for sale. The site also gave a list of all the houses in the neighborhood, when they were last sold, and who was the seller and the buyer. I discovered that her house had been sold in May 2013. It was possible I was just over a year too late! She and her husband were listed as the sellers, but estate was written after his name. I called the real estate agent who sold the house. He told me that this now 93-year-old aunt was still very much alive, and he gave me her phone number. I called the number and was able to talk to her!

It turned out that my son, John, lived only 40 minutes away from Aunt Betty! He immediately made arrangements to meet her. Steven and I flew out to Virginia at the end of July, and John took us over to meet her as well.  She shared stories and pictures with me that I would never have otherwise known or seen. Since she was 93 years old, she had a caregiver, Ingrid Graham, who was absolutely wonderful. Ingrid explained that after Betty sold her house, they moved into an apartment that included amenities that Betty couldn’t take advantage of, so they moved again. The real estate agent would not have known of this second move except Betty continued to get a gas bill for the house she sold. About a week before I called the agent, her caregiver had written the real estate agent a letter requesting his assistance in resolving the gas bill, and the letter had her new address and phone number. Thus the gas bill mix-up was part of the miracle of finding Aunt Betty! This trip to meet her was the highlight of 2014 for me.

Here is a picture of Betty when she was younger. The picture was taken by my husband’s father:

Scan0019 (1)

And here is a picture of my husband Steve, Betty, and me that was taken last summer.

Sadly Betty died in December 2014. My husband flew out to Ohio to attend her funeral, but I was recovering from surgery and couldn’t travel. Ingrid planned a memorial service for her in April because there were others who wanted to attend the funeral in December but couldn’t. I was very grateful to be able to attend the memorial service yesterday and reconnect with Ingrid and others who were part of Aunt Betty’s life.

Elizabeth Bender memorial service

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Now I’ll share the factoring information for the number 465.

465 = 1 + 2 + 3 + . . . + 28 + 29 + 30, so it is a triangular number represented by (30 x 31)/2.

465 is formed by three consecutive digits so it can be evenly divided by 3. It is not divisible by 9 because the middle digit of the three consecutive digits, 5, is not a multiple of 3.

  • 465 is a composite number.
  • Prime factorization: 465 = 3 x 5 x 31
  • 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 465 has exactly 8 factors.
  • Factors of 465: 1, 3, 5, 15, 31, 93, 155, 465
  • Factor pairs: 465 = 1 x 465, 3 x 155, 5 x 93, or 15 x 31
  • 465 has no square factors that allow its square root to be simplified. √465 ≈ 21.5638