factor pairs

1401 Roasting Over an Open Fire

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I went camping last week. My family roasted hotdogs. Some people refer to them as mystery meat. Others roasted marshmallows. I was surprised to learn that almost all brands of marshmallows have blue dye in them.  I’m told that without that blue dye the marshmallows will lose their whiteness as they sit on store shelves. Why they have to be that white is a mystery to me.

Here’s a mystery level puzzle for you to solve. It looks a lot like the utensil that was used to roast the hotdogs and marshmallows.

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

Now I’ll tell you something about the number 1401:

  • 1401 is a composite number.
  • Prime factorization: 1401 = 3 × 467.
  • 1401 has no exponents greater than 1 in its prime factorization, so √1401 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 1401 has exactly 4 factors.
  • The factors of 1401 are outlined with their factor pair partners in the graphic below.

1401 is the difference of two squares in two different ways. Can you figure out what those ways are?

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

 

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.

 

 

1391 and Level 1

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Many of the clues in this puzzle have double digits. If you know why they do, then you can find all the factors and solve this puzzle!

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

Here’s some information about the number 1391:

  • 1391 is a composite number.
  • Prime factorization: 1391 = 13 × 107
  • 1391 has no exponents greater than 1 in its prime factorization, so √1391 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 1391 has exactly 4 factors.
  • The factors of 1391 are outlined with their factor pair partners in the graphic below.

1391 is the hypotenuse of a Pythagorean triple:
535-1284-1391 which is (5-12-13) times 107