Prime Factorization of the Hundred Numbers up to 1400

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Almost one-third of the numbers from 1301 to 1400 have 4 factors. Only 1/5 of the numbers have 8 factors.

Since 1/3 is significantly bigger than 1/5, the amount of factors for these numbers wouldn’t make a very exciting horse race. Here is the breakdown:

  • 11 numbers had 2 factors
  • 1 number had 3 factors
  • 32 numbers had 4 factors
  • 7 numbers had 6 factors
  • 20 numbers had 8 factors
  • 2 numbers had 10 factors
  • 13 numbers had 12 factors
  • 4 numbers had 16 factors
  • 1 number had 18 factors
  • 2 numbers had 20 factors
  • 5 numbers had 24 factors
  • 1 number had 28 factors
  • 1 number had 32 factors

The rosy looking numbers have square roots that can be simplified, and that is only 37% of the numbers listed.

You may not expect it, but 1400 is one of the numbers with 24 factors. Let me tell you a little bit about 1400 and why it has so many factors:

  • 1400 is a composite number.
  • Prime factorization: 1400 = 2 × 2 × 2 × 5 × 5 × 7, which can be written 1400 = 2³ × 5² × 7
  • 1400 has at least one exponent greater than 1 in its prime factorization so √1400 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1400 = (√100)(√14) = 10√14
  • The exponents in the prime factorization are 3, 2, and 1. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1400 has exactly 24 factors.
  • The factors of 1400 are outlined with their factor pair partners in the graphic below.

1400 is the hypotenuse of TWO Pythagorean triples:
392-1344-1400 which is (7-24-25) times 56
840-1120-1400 which is (3-4-5) times 280

1399 and Level 6

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The eligible common factors of 48 and 72 are 6, 8, and 12. The common factors for 10 and 30 are 5 and 10.  Don’t guess and check the possibilities! Can you figure out the logic needed to start this puzzle?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s a little information about the number 1399:

  • 1399 is a prime number.
  • Prime factorization: 1399 is prime.
  • 1399 has no exponents greater than 1 in its prime factorization, so √1399 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1399 has exactly 2 factors.
  • The factors of 1399 are outlined with their factor pair partners in the graphic below.

How do we know that 1399 is a prime number? If 1399 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1399. Since 1399 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1399 is a prime number.

1399 is the difference of two squares:
700² – 699² = 1399

 

1398 and Level 5

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You might find this puzzle to be a little tricky, but if you always use logic before you write any of the factors, you should succeed!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here is some information about the number 1398:

  • 1398 is a composite number.
  • Prime factorization: 1398 = 2 × 3 × 233
  • 1398 has no exponents greater than 1 in its prime factorization, so √1398 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1398 has exactly 8 factors.
  • The factors of 1398 are outlined with their factor pair partners in the graphic below.

1398 is the hypotenuse of a Pythagorean triple:
630-1248-1398 which is 6 times (105-208-233)

1397 and Level 4

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I bet you know enough multiplication facts to get this puzzle started. Once you’ve started it, you might as well finish it. You will feel so clever when you do!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll write a little bit about the puzzle number, 1397:

  • 1397 is a composite number.
  • Prime factorization: 1397 = 11 × 127
  • 1397 has no exponents greater than 1 in its prime factorization, so √1397 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1397 has exactly 4 factors.
  • The factors of 1397 are outlined with their factor pair partners in the graphic below.

1397 is the difference of two squares two different ways:
699² – 698² = 1397
69² – 58² = 1397

31 Flavors of 1396

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The first 52 triangular numbers are 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378.

OEIS.org informs us that 1396 can be written as the sum of three triangular numbers in 31 different ways. It is the smallest number that can make that claim!

That 31st way is written with three consecutive triangular numbers, 435, 465, and 496, which are the 29th, 30th, and 31st triangular numbers respectively. That fact makes 1396 the 31st Centered Triangular Number as well!

That is, at least, 1396 is the 31st number on the list. You can also calculate it using this formula: [3(30²) + 3(30) + 2]/2 = 1396

Here’s more about the number 1396:

  • 1396 is a composite number.
  • Prime factorization: 1396 = 2 × 2 × 349, which can be written 1396 = 2² × 349
  • 1396 has at least one exponent greater than 1 in its prime factorization so √1396 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1396 = (√4)(√349) = 2√349
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1396 has exactly 6 factors.
  • The factors of 1396 are outlined with their factor pair partners in the graphic below.

1396 is the sum of two squares:
36² + 10² = 1396

1396 is the hypotenuse of a Pythagorean triple:
720-1196-1396 calculated from 2(36)(10), 36² – 10², 36² + 10²

1395 and Level 2

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1391 is the 22nd Friedman number, and there are TWO reasons why!

See! Factoring numbers can be such an exciting adventure! Can you find the factors for this puzzle?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403\

Here’s more about the number 1395:

  • 1395 is a composite number.
  • Prime factorization: 1395 = 3 × 3 × 5 × 31, which can be written 1395 = 3² × 5 × 31
  • 1395 has at least one exponent greater than 1 in its prime factorization so √1395 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1395 = (√9)(√155) = 3√155
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1395 has exactly 12 factors.
  • The factors of 1395 are outlined with their factor pair partners in the graphic below.

You can see the reasons 1395 is the 22nd Friedman numbers in these factor pairs:
15 × 93 = 1395
45 × 31 = 5×9×31 = 1395, that one uses the digits in reverse order!

1395 is also the hypotenuse of a Pythagorean triple:
837-1116-1395 which is (3-4-5) times 279

 

1394 and Level 2

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The factors and most of the products are missing from this multiplication table, and the ones that are there aren’t in there usual places. Can you figure out where everything goes?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll write a little bit about the puzzle number, 1394:

  • 1394 is a composite number.
  • Prime factorization: 1394 = 2 × 17 × 41
  • 1394 has no exponents greater than 1 in its prime factorization, so √1394 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1394 has exactly 8 factors.
  • The factors of 1394 are outlined with their factor pair partners in the graphic below.

1394 is the hypotenuse of FOUR Pythagorean triples:
306-1360-1394 which is (9-40-41) times 34
370-1344-1394 which is 2 times (185-672-697)
656-1230-1394 which is (8-15-17) times 82
910-1056-1394 which is 2 times (455-528-697)

1393 DNA Shared Matches

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Ancestry.com gave my husband a list of his 50 top matches of DNA from their database. For each match they found, I could click on a button that would reveal any matches that my husband shared with that match. Some of his matches didn’t share any other match with him. Sometimes a couple of their shared matches didn’t make his list of top 50 matches. I made a table of his shared matches. It was pretty big so I made a smaller table that only includes people in his top 50 who have at least one shared match with him AND a second or third cousin.

I purposely cut off people’s names for privacy reasons, but anyone who shares DNA with my husband and the others in the table should still be able to figure out who’s who.

Ancestry explains that a 2nd cousin could actually be a great aunt or a 1st cousin twice removed. The 2nd cousin would have 5 to 6 degrees of separation from my husband, a 3rd cousin would have 6 to 10 degrees of separation, and a 4th cousin would have 6 to 12 degrees of separation, but most likely 10.

DNA does NOT “share and share alike”. Every person gets half of his DNA from his mother and a half from his father, but the half given from each parent can vary from child to child. I noticed that some of my husband’s matches might be siblings with the same surname, but their shared matches were not always the same. Thus, it can definitely be worth it to have more than one family member take the DNA test.

I made this chart to see if it could help me determine who might be my husband’s maternal cousins versus his paternal cousins. I don’t think I completely succeeded. The same DNA might not be the DNA in shared matches. For example, ab, bc, and ac each share a letter of the alphabet with each other, but it is not the same letter of the alphabet. Since both sides of my husband’s family had many siblings and cousins and settled in the Cleveland, Ohio area 100 years ago or more, it seems possible that some of his relatives listed on the chart are actually related to BOTH his father and his mother, but more distantly than 4th cousin on either side.

A positive from making the chart is that I have verified that all the people with x’s in the lower right corner are closely related to each other. The chart says they are also all related to Benjam, but none of them have any idea how.

Like so much of genealogy research, one answer will produce more questions. It becomes such a fascinating puzzle!

Since this is my 1393rd post, I’ll write a little bit about that number:

  • 1393 is a composite number.
  • Prime factorization: 1393 = 7 × 199
  • 1393 has no exponents greater than 1 in its prime factorization, so √1393 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1393 has exactly 4 factors.
  • The factors of 1393 are outlined with their factor pair partners in the graphic below.

Since both of its factor pairs have odd numbers in it, I know that 1393 can be written as the difference of two squares in two ways:
697² – 696² = 1393
103² – 96² = 1393

 

DNA and Big Brother

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When my husband was a little baby, his dad filled out the genealogy section in his baby book in his beautiful, distinct handwriting:

Even though most of the pages are blank, my husband has always cherished that book, and it has been extremely helpful in finding many other of his ancestors.

From additional research, we have learned that my husband’s grandfather, Frank Kovach, was born Kovács Ferenc in Szürte, Ung County, Hungary. That little town has had several border changes and is now part of Ukraine, but still only about eight miles from the Hungarian border. You can see a map showing the location of Szürte in a post I wrote a couple of years ago. Ferenc (Frank) was born 13 June 1883 to Kovács Péter and Péntek Mária – that’s their names in Hungarian name order. The baby book gives their names in English name order. My husband remembers his grandfather, Frank, vividly. He died 10 June 1968 in Ontario, San Bernardino, California.

Many years ago when I tried to figure out Frank’s place of birth, I found three other people whose parents had the same names as his parents. Could they be Frank’s siblings? Could the two boys be his big brothers? (You will need to be logged into FamilySearch.org and Ancestry.com to see most of the links I’ve included in this post.)

  1. Julia Kovach (Kovács Juliánna) was born 12 Apr 1882 in Hungary (both of her parents were born in Ung County, Hungary!). She died 15 Jun 1940 in Cleveland, Ohio. Maybe Frank was also born in Ung County, I excitedly thought! Several years later I found a death record for one of Frank’s sons that gave the specific town in Ung county where Frank was born. Still years after that I found Frank’s petition for naturalization also confirming it.
  2. Steven Kovach (Kovács István) was born about 1874 in Hungary. He married Julia Csengeri on 22 Sept 1901 in New York.

    He MAY have died seventeen years later on 11 Dec 1918 in Union, Washington, Pennsylvania, but buried in Cleveland, Ohio.  The father on that death certificate was Pete Kovacs and the mother was Mary Pantik. The certificate says he is married, but there was no place to write his wife’s name on it. The informant was Steve Kovach, which just happens to be Frank’s sister’s (Julia Kovach) husband’s name, so his brother-in-law might have actually been the informant. Julia and Steve lived in Cleveland, and the deceased, Steve, was buried in Cleveland even though he died in Pennsylvania.
  3. John Kovacs (Kovács János) was born 23 Jan 1870 in Hungary. He died 29 Oct 1943 in Cleveland. To fully appreciate the information for John, we need to look at his and his wife’s death certificates side by side.

Notice that the address for both John and Veronica is 9012 Cumberland, so that helps to establish that they were husband and wife even though the spellings of their last names are not exactly the same. This is important since there were MANY men named John Kovach in Cleveland. The couple’s shared tombstone confirms the dates given above. On Veronica’s death certificate, her father is listed as John Daniels and the informant is Dale Kovats. Further research establishes that Dale is John and Veronica’s son, and the 1940 census shows Dale and his wife, Rose at the bottom of the page, and their daughter and some of Rose’s relatives on the top of the next page. Dale is the key to this puzzle because Dale has a descendant who is a 3rd to 4th cousin DNA match to my husband! That means that John Kovacs is indeed Frank’s big brother, and I am in tears as I am finally able to positively make that statement.

Ancestry.com explains “Our analysis of your DNA predicts that this person you match with is probably your third cousin. The exact relationship however could vary. It could be a second cousin once removed, or perhaps a fourth cousin. While there may be some statistical variation in our prediction, it’s likely to be a third cousin type of relationship—which are separated by eight degrees or eight people. However, the relationship could range from six to ten degrees of separation.” (bold print added)

My husband, Steven, and this DNA match are separated by seven degrees.

Was big brother John also born in Szürte? It seems likely, but he may have also been born about 3 miles away in Kholmetz where a 4th-6th cousin DNA shared match traces her ancestry. If only I could get into the Szürte Reformed Church records and Kholmetz records to look for a Kovács János (John Kovacs) born on 23 Jan 1870 as well as the records for the others and certainly a few more siblings as well!

 

1392 and Pythagorean Triples

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1392 is the hypotenuse of ONE Pythagorean triple, 960-1008-1392.

However, 1392 is the leg of so many Pythagorean triples, that it is possible I haven’t listed them all in this graphic:


Why is it the hypotenuse only once, but it is a leg so many times?

Because of its factors!

  • 1392 is a composite number.
  • Prime factorization: 1392 = 2 × 2 × 2 × 2 × 3 × 29, which can be written 1392 = 2⁴ × 3 × 29
  • 1392 has at least one exponent greater than 1 in its prime factorization so √1392 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1392 = (√16)(√87) = 4√87
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1392 has exactly 20 factors.
  • The factors of 1392 are outlined with their factor pair partners in the graphic below.

1392 has only one prime factor that leaves a remainder of one when it is divided by four. That factor is 29. It makes 960-1008-1392 simply (20-21-29) times 48. Easy Peasy.

ONE of the reasons it is a leg so many times is because several of its factors are in primitive Pythagorean triples, and multiplying those triples by that factor’s factor pair gives us a triple with 1392 as a leg:

  • (3-4-5) times 464 is (1392-1856-2320)
  • (3-4-5) times 348 is (1044-1392-1740)
  • (8-15-17) times 174 is (1392-2610-2958)
  • (5-12-13) times 116
  • (12-35-37) times 116, and so on

Another reason is every Pythagorean triple can be written in this form 2ab, a²-b², a²+b², and 1392 = 2(696)(1) or 2(348)(2) or 2(232)(3) or 2(174)(4) and so on.

The last reason is that since 1392 has six factor pairs in which both factors are even, it can be written as a²-b²: (The average of the two numbers in the factor pair gives us the first number to be squared. Subtract the second number from it to get the second number to be squared.)

  • 696 and 2 give us 349² – 347² = 1392
  • 348 and 4 give us 176² – 172² = 1392
  • 232 and 6 give us 119² – 113² = 1392
  • 174 and 8 give us 91² – 83² = 1392
  • 116 and 12 give us 64² – 52² = 1392
  • 58 and 24 give us 41² – 17² = 1392

Some of the triples can be found by more than one of the processes listed above. It can be very confusing to keep track of them all. That is why I usually only write when a number is the hypotenuse of a triple and not when it is a leg.