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

—————————

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

 

464 and Level 6

464 is the hypotenuse of Pythagorean triple 320-336-464. Can you figure out what is the greatest common factor of those three numbers? Hint: it has to be an even factor of 320 because that is the smallest of those three even numbers.

The logic needed to begin this Level 6 puzzle shouldn’t be too difficult to discover.

464 Puzzle

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

—————————————————————————————————

If I wanted to simplify √464, I would first notice that its last two digits, 64, are divisible by 4, so 464 also is divisible by 4. I would make a little cake like this:

464 divided by 4

464 ÷ 4 = 116. Guess what? 116 is also divisible by 4 because 16 is divisible by 4. I would make another layer for my cake like this:

464 two layer cake

29 is a prime number so my cake is finished. Now to simplify √464, I would just take the square root of everything on the outside of the cake and multiply them together.

√464 = (√4)(√4)(√29) = 4√29

  • 464 is a composite number.
  • Prime factorization: 464 = 2 x 2 x 2 x 2 x 29, which can be written 464 = (2^4) x 29
  • 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 464 has exactly 10 factors.
  • Factors of 464: 1, 2, 4, 8, 16, 29, 58, 116, 232, 464
  • Factor pairs: 464 = 1 x 464, 2 x 232, 4 x 116, 8 x 58, or 16 x 29
  • Taking the factor pair with the largest square number factor, we get √464 = (√16)(√29) = 4√29 ≈ 21.5407

—————————————————————————————————

464 Logic

463 and Level 5

463 is the sum of consecutive primes, too! Check the comments to see if any of my readers finds out what those consecutive primes are.

This Level 5 puzzle might be a little harder than usual. If you’ve solved a Level 5 puzzle before, see if you can meet this challenge!

463 Puzzle

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

—————————————————————————————————

  • 463 is a prime number.
  • Prime factorization: 463 is prime.
  • The exponent of prime number 463 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 463 has exactly 2 factors.
  • Factors of 463: 1, 463
  • Factor pairs: 463 = 1 x 463
  • 463 has no square factors that allow its square root to be simplified. √463 ≈ 21.5174

How do we know that 463 is a prime number? If 463 were not a prime number, then it would be divisible by at least one prime number less than or equal to √463 ≈ 21.5174. Since 463 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 463 is a prime number.

—————————————————————————————————

463 Logic

 

 

462 and Level 4

462 is the sum of consecutive prime numbers two different ways. Check the comments to see what those ways are.

Divisibility tricks:

  • 462 is even, so it is divisible by 2.
  • The sum of the odd numbered digits, 4 + 2 is 6, which is the 2nd digit, so 462 is divisible by 11.
  • Since both of those 6’s above are divisible by 3, then 462 is divisible by 3.
  • Separate the last digit from the rest and double it. 462 → 46 and 2; Doubling 2, gives us 4. Now subtract that 4 from the remaining digits: 46 – 4 = 42 which is divisible by 7, so 462 is divisible by 7.

Since 462 = 21 × 22, we know that it is two times the 21st triangular number, and it is the sum of the first 21 even numbers.

  • 2(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21) = 462
  • 2 + 4 +  6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 = 462

462 Puzzle

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

—————————————————————————————————

  • 462 is a composite number.
  • Prime factorization: 462 = 2 x 3 x 7 x 11
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 462 has exactly 16 factors.
  • Factors of 462: 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462
  • Factor pairs: 462 = 1 x 462, 2 x 231, 3 x 154, 6 x 77, 7 x 66, 11 x 42, 14 x 33, or 21 x 22
  • 462 has no square factors that allow its square root to be simplified. √462 ≈ 21.4942

—————————————————————————————————

462 Logic

461 and Level 3

461 = 19² + 10², and it is the hypotenuse in this primitive Pythagorean triple: 261-380-461.

461 Puzzle

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

—————————————————————————————————

  • 461 is a prime number.
  • Prime factorization: 461 is prime.
  • The exponent of prime number 461 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 461 has exactly 2 factors.
  • Factors of 461: 1, 461
  • Factor pairs: 461 = 1 x 461
  • 461 has no square factors that allow its square root to be simplified. √461 ≈ 21.4709

How do we know that 461 is a prime number? If 461 were not a prime number, then it would be divisible by at least one prime number less than or equal to √461 ≈ 21.4709. Since 461 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 461 is a prime number.

—————————————————————————————————

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.

461 Factors

460 Happy Birthday, Tim!

460 is the sum of consecutive prime numbers. Check the comments because one of my readers was able to find what those consecutive primes are.

Happy birthday to my son, Tim. I have two different cakes for you in this post. A cake puzzle and a simplified square root that uses the cake method that I’ve modified.

Happy birthday, Tim

This puzzle will be included in an excel file of puzzles 12 Factors 2015-04-20.

—————————————————————————————————

When we simplify square roots, we want to do as few divisions as possible. Since 60 can be evenly divided by perfect square 4, we know that 460 is also divisible by 4. Let’s use that fact to find its square root:

460 one layer cake

The quotient, 115, may be too large for us to know if it has any square factors. Since it isn’t divisible by 4, 9, or 25, let’s make a second layer to our cake as we divide it by its largest prime factor, 5.

460 two layer cake

Since the new quotient, 23, is a prime number, let’s revert back to the previous cake and take the square root of everything on the outside of the one layer cake: √460 = (√4)(√115) = 2√115.

  • 460 is a composite number.
  • Prime factorization: 460 = 2 x 2 x 5 x 23, which can be written 460 = (2^2) x 5 x 23
  • 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 460 has exactly 12 factors.
  • Factors of 460: 1, 2, 4, 5, 10, 20, 23, 46, 92, 115, 230, 460
  • Factor pairs: 460 = 1 x 460, 2 x 230, 4 x 115, 5 x 92, 10 x 46, or 20 x 23
  • Taking the factor pair with the largest square number factor, we get √460 = (√4)(√115) = 2√115 ≈ 21.4476

—————————————————————————————————

460 Factors

Here’s the order the factors were found:

460 Logic

459 and Level 2

459 is the hypotenuse of this Pythagorean triple: 216-405-459.

What is the greatest common factor of those three numbers?

The GCF has to be a factor of the smallest number, 216, and it has to be an odd number because at least one of the other numbers is odd. Let’s factor out the even factors of 216 to find its greatest odd factor:

  • 216 can be evenly divided by 4 because the last two digits form a multiple of 4.
  • It can also be evenly divided by 8 because 16 is a multiple of 8 and the 3rd from the right digit is even.
  • 216 ÷ 8 = 27.
  • Check to see if the other two numbers in the triple are divisible by 27, and you will see that 27 is the GCF of 216-405-459.

459 Puzzle

To solve this puzzle ask yourself:

What is a common factor of 6, 14, 10, and 16? What about 3, 18, 6, and 30? And what is a common factor of 9, 36, 45, 63, 54, 90, 72? In each case, the common factor has to be a factor of the smallest number on the list, and if any of the numbers on the list are odd, it has to be an odd number. (For level 1 and level 2 puzzles, that factor will oftrn be the greatest common factor of all the numbers in a particular row or column.)

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

—————————————————————————————————

459 cannot be evenly divided by 100 or by 4, but it is divisible by 9. To find it square root, let’s first divide 459 by 9:

459 divided by 9

 

The quotient, 51, is small enough that we can recognize that it cannot be evenly divided by any square number less than it. Thus we take the square root of everything on the outside of the cake and get √459 = (√9)(√51) = 3√51.

  • 459 is a composite number.
  • Prime factorization: 459 = 3 x 3 x 3 x 17, which can be written 459 = (3^3) x 17
  • 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 459 has exactly 8 factors.
  • Factors of 459: 1, 3, 9, 17, 27, 51, 153, 459
  • Factor pairs: 459 = 1 x 459, 3 x 153, 9 x 51, or 17 x 27
  • Taking the factor pair with the largest square number factor, we get √459 = (√9)(√51) = 3√51 ≈ 21.4243

—————————————————————————————————

459 Factors

458 and Level 1

458 = (13^2) + (17^2). It is the hypotenuse of this Pythagorean triple: 120-442-458.

458 Puzzle

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

—————————————————————————————————

  • 458 is a composite number.
  • Prime factorization: 458 = 2 x 229
  • 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 458 has exactly 4 factors.
  • Factors of 458: 1, 2, 229, 458
  • Factor pairs: 458 = 1 x 458 or 2 x 229
  • 458 has no square factors that allow its square root to be simplified. √458 ≈ 21.4009

—————————————————————————————————

458 Factors

457 A Pythagorean Triple Logic Puzzle

457 = 4² + 21², and it is the hypotenuse of the primitive Pythagorean triple 168-425-457. Also, 457 is the sum of some consecutive prime numbers. One of my readers posted those primes in the comments.

A long time ago I decided that Pythagorean triples could make a great logic puzzle, so I created one. You can see it directly underneath the following directions:

This puzzle is NOT drawn to scale. Angles that are marked as right angles are 90 degrees, but any angle that looks like a 45 degree angle, isn’t 45 degrees. Lines that look parallel are NOT parallel. Shorter looking line segments may actually be longer than longer looking line segments. Most rules of geometry do not apply here: in fact non-adjacent triangles in the drawing might actually overlap.

No geometry is needed to solve this puzzle. All that is needed is the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.

If any of these directions are not clear, let me know in the comments. I will NOT be publishing the solution to this puzzle, but I will allow anyone who desires to put any or all of the missing values in the comments. Also, the comments will help me determine if I should publish another puzzle like this one.

Good Luck!

457 Puzzle

Sorted Triples

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

—————————————————————————————————

  • 457 is a prime number.
  • Prime factorization: 457 is prime.
  • The exponent of prime number 457 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 457 has exactly 2 factors.
  • Factors of 457: 1, 457
  • Factor pairs: 457 = 1 x 457
  • 457 has no square factors that allow its square root to be simplified. √457 ≈ 21.3776

How do we know that 457 is a prime number? If 457 were not a prime number, then it would be divisible by at least one prime number less than or equal to √457 ≈ 21.3776. Since 457 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 457 is a prime number.

456 An Inchworm Measuring Marigolds

456 is the sum of consecutive prime numbers in two different ways. One of my readers listed those ways in the comments. The factors of 456 are at the end of the post.

Inchworm, inchworm,
Measuring the marigolds
You and your arithmetic will probably go far.

Two plus two is four
Four plus four is eight
Eight and eight is sixteen
Sixteen and sixteen is thirty-two.

Inchworm, inchworm,
Measuring the marigolds
Seems to me you’d stop and see
How beautiful they are.

Today I taught a class of three year olds about being thankful for birds, insects, and creeping things. To keep their attention, I used a variety of stories, riddles, books, and games. I also sang a few songs including this one about an inchworm who is very good at arithmetic. I think preschool children can still enjoy songs like this even if they don’t understand everything the song is about or even if they are wiggling as much as an inchworm while they listen to it. Here is the song sung by Danny Kaye from the movie Hans Christian Andersen:

———————————————————————————————————

Now for the number 456. The last two digits can be evenly divided by four, so the entire number is divisible by four. Also since it is formed from three consecutive numbers, it is divisible by 3. However since the number in the middle of those consecutive numbers is not 3, 6, 9 or another multiple of 3, we know that 456 is NOT divisible by 9.

Because it is divisible by four, we will use that fact first to determine how to reduce its square root.

456 divided by 4

456 ÷ 4 = 114. Notice that 114 is even, but 14 can’t be evenly divided by 4, so 114 cannot be either. Also notice that 114 is still divisible by 3. If we’re not sure whether or not 114 has any square factors, we are less likely to make a mistake if we divide it by 6 once, instead of by 2 and then by 3.

114 divided by 6

114 ÷ 6 = 19, a prime number, and we are certain there were no other square factors. Since we know 19 x 6 = 114, let’s backtrack a little and go back to that original one layer cake:

456 divided by 4

Take the square root of everything on the outside of the cake and get √456 = (√4)(√114) = 2√114

———————————————————————————————————

  • 456 is a composite number.
  • Prime factorization: 456 = 2 x 2 x 2 x 3 x 19, which can be written 456 = (2^3) x 3 x 19
  • 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 456 has exactly 16 factors.
  • Factors of 456: 1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456
  • Factor pairs: 456 = 1 x 456, 2 x 228, 3 x 152, 4 x 114, 6 x 76, 8 x 57, 12 x 38, or 19 x 24
  • Taking the factor pair with the largest square number factor, we get √456 = (√4)(√114) = 2√114 ≈ 21.3542

———————————————————————————————————

Picture credits: Inchworm and ruler: http://www.kindergartenkindergarten.com/2012/06/problem-solving-measurement.html;