Number

1343 Level TWO Puzzles Only

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Level 2 puzzles are much more interesting-looking than level 1 puzzles, but they are still relatively easy for beginners to solve. I decided to put all the level 2 puzzles from 2018 into one collection. You can use the image I put at the top of the post to work on solving them or you can find the complete collection at Level 2’s from 2018.

Now I’ll write a little bit about the number 1343:

  • 1343 is a composite number.
  • Prime factorization: 1343 = 17 × 79
  • 1343 has no exponents greater than 1 in its prime factorization, so √1343 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 1343 has exactly 4 factors.
  • The factors of 1343 are outlined with their factor pairs in the graphic below.

1343 is the hypotenuse of a Pythagorean triple:
632-1185-1343 which is (8-15-17) times 79

OEIS.org informs us that 1343 is 16 numbers away from the closest prime number, and it is the smallest number that can make that claim.

1342 Level One Puzzles Only

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My son requested I made some pages using only level 1 puzzles for my granddaughter to do. You can see and use the first page above, but I’ve also compiled ALL the level 1 puzzles from 2018 and put them in one excel file: Level 1’s from 2018

Those puzzles will give beginners lots of practice and help them learn the multiplication table.

Now I’ll tell you a little bit about the number 1342:

  • 1342 is a composite number.
  • Prime factorization: 1342 = 2 × 11 × 61
  • 1342 has no exponents greater than 1 in its prime factorization, so √1342 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 1342 has exactly 8 factors.
  • The factors of 1342 are outlined with their factor pairs in the graphic below.

1342 is the hypotenuse of a Pythagorean triple:
242-1320-1342 which is 22 times (11-60-61)

And (11-60-61) times 122 is a triple where 1342 is a leg:
1342-7320-7442. Pretty cool!

The last prime number was 1327 which was 15 numbers ago. The next prime number won’t be until 1361 which is 19 numbers away. Wow! That means that 1342 is 15 numbers away from the CLOSEST prime number, a new record. Thank you OEIS.org for alerting me to that fact.

Facts about and Factors of 2019

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Here’s a countdown you can use to ring in the New Year:

Countdown to 2019

make science GIFs like this at MakeaGif

2019 is the sum of consecutive numbers three different ways:
1009 + 1010 = 2019
672 + 673 + 674 = 2019
334 + 335 + 336 + 337 + 338 + 339 = 2019

There is one way that 2019 is the sum of consecutive odd numbers:
671 + 673 + 675 = 2019

2019 is the difference of two squares two different ways:
338² – 335² = 2019
1010² – 1009² = 2019

2019 is the sum of three squares nine different ways:
43² + 13² + 1² = 2019
43² + 11² + 7² = 2019
41² + 17² + 7² = 2019
41² + 13² + 13² = 2019
37² + 25² + 5² = 2019
37² + 23² + 11² = 2019
37² + 19² + 17² = 2019
35² + 25² + 13² = 2019
31² + 23² + 23² = 2019

2019 is the hypotenuse of a Pythagorean triple:
1155-1656-2019 so 1155² + 1656² = 2019²

2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2¹ + 2⁰ = 2019

2019 is a palindrome in a couple of bases:
It’s 5B5 in BASE 19 (B is 11 base 10) because 5(19²) + 11(19) + 5(1) = 2019,
and 3C3 in BASE 24 (C is 12 base 10) because 3(24²) + 12(24) + 3(1) = 2019

Every year has factors that often catch people by surprise. Today I would like to give you my predictions for the factors of 2019:
2019 will have four positive factors: 1, 3, 673, and 2019
However, 2019 will also have four negative factors: -1, -3, -673, and -2019

Which factors, positive or negative, will be your focus in the coming year?

Finally, I’ll share some mathematics-related 2019 and New Year tweets that I’ve seen on twitter. Some of these tweets have links that contain even more facts about the number 2019.

That tweet inspired me to make my own 3 × 3 Magic Square where every number is different but every row, column and diagonal totals the same number:

But because 2019 is divisible by 3, it can also be a magic sum of 3 × 3 magic square:

5

5

And finally, here is my contribution to 2019 twitter:

 
 
 
 
 
 
 
 

1341 Not a Pretty Picture

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In real mysteries, sometimes clues are close together, sometimes they are far apart. They don’t often make a pretty picture. Can you use the clues and logic to solve this mystery?

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here is some information about the number 1341:

  • 1341 is a composite number.
  • Prime factorization: 1341 = 3 × 3 × 149, which can be written 1341 = 3² × 149
  • 1341 has at least one exponent greater than 1 in its prime factorization so √1341 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1341 = (√9)(√149) = 3√149
  • 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 1341 has exactly 6 factors.
  • The factors of 1341 are outlined with their factor pairs in the graphic below.

1341 is the sum of two squares:
30² + 21² = 1341

1341 is the hypotenuse of a Pythagorean triple:
459-1260-1341 which is 9 times (51-140-149) and
can also be calculated from 30² – 21², 2(30)(21), 30² + 21²

1340 Mystery Level

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Which two of these clues will use both of the 1’s? 10, 12, 9, or 7. I’ll never tell. You will have to solve that mystery yourself!

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here are a few facts about the number 1340:

  • 1340 is a composite number.
  • Prime factorization: 1340 = 2 × 2 × 5 × 67, which can be written 1340 = 2² × 5 × 67
  • 1340 has at least one exponent greater than 1 in its prime factorization so √1340 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1340 = (√4)(√335) = 2√335
  • 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 1340 has exactly 12 factors.
  • The factors of 1340 are outlined with their factor pairs in the graphic below.

1340 is the hypotenuse of a Pythagorean triple:
804-1072-1340 which is (3-4-5) times 268

 

1339 Written Eligible Factor Pairs Can Reveal the Needed Logic

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The logic needed for this particular level 6 puzzle is VERY complicated so I wrote out the eligible factor pairs of all the clues for you.

Or you can start from scratch:

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here are a few facts about the puzzle number, 1339:

  • 1339 is a composite number.
  • Prime factorization: 1339 = 13 × 103
  • 1339 has no exponents greater than 1 in its prime factorization, so √1339 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 1339 has exactly 4 factors.
  • The factors of 1339 are outlined with their factor pairs in the graphic below.

1339 is the hypotenuse of a Pythagorean triple:
515-1236-1339 which is (5-12-13) times 103

1338 and Level 5

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Finding the most logical place to start a level 5 puzzle is only a little bit trickier than for a level 4 puzzle. Will you figure it out and not be tricked?

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here is some information about the number 1338:

  • 1338 is a composite number.
  • Prime factorization: 1338 = 2 × 3 × 223
  • 1338 has no exponents greater than 1 in its prime factorization, so √1338 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 1338 has exactly 8 factors.
  • The factors of 1338 are outlined with their factor pairs in the graphic below.

1338 is in a couple of Pythagorean triples:
1338-447560-447562 calculated from 2(669)(1), 669² – 1², 669² + 1² and
1338-49720-49738 calculated from 2(223)(3), 223² – 3², 223² + 3²

1337 and Level 4

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If you can solve a level 3 puzzle, give this one a try. You will have to locate the best clues to use in the beginning and then what is the most logical clue to use next over and over, but I think you can do that!

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here is some information about the puzzle number, 1337:

  • 1337 is a composite number.
  • Prime factorization: 1337 = 7 × 191
  • 1337 has no exponents greater than 1 in its prime factorization, so √1337 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 1337 has exactly 4 factors.
  • The factors of 1337 are outlined with their factor pairs in the graphic below.

1337 is the difference of two squares two ways:
99² – 92² = 1337
669² – 668² = 1337

1336 and Level 3

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To solve this level 3 puzzle, start with the clue at the top of the grid, write the factors from 1 to 10 in the appropriate places, then do the same for each factor going down the puzzle row by row.

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here’s some information about the puzzle number, 1336:

  • 1336 is a composite number.
  • Prime factorization: 1336 = 2 × 2 × 2 × 167, which can be written 1336 = 2³ × 167
  • 1336 has at least one exponent greater than 1 in its prime factorization so √1336 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1336 = (√4)(√334) = 2√334
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1336 has exactly 8 factors.
  • The factors of 1336 are outlined with their factor pairs in the graphic below.

1336 is the difference of two squares two ways:
335² – 333² = 1336
169² – 165² = 1336

1335 is a Pentagonal Number

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1335 is the 30th Pentagonal Number. Here’s a graphic to help you visualize that fact:

Here are some more facts about the number 1335:

  • 1335 is a composite number.
  • Prime factorization: 1335 = 3 × 5 × 89
  • 1335 has no exponents greater than 1 in its prime factorization, so √1335 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 1335 has exactly 8 factors.
  • The factors of 1335 are outlined with their factor pairs in the graphic below.

1335 is also the hypotenuse of FOUR Pythagorean triples:
252-1311-1335 which is 3 times (84-437-445)
585-1200-1335 which is 15 times (39-80-89)
609-1188-1335 which is 3 times (203-396-445)
801-1068-1335 which is (3-4-5) times 267