Factors

1345 Level Three Puzzles Only

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To solve a level 3 puzzle, start with a clue in the top row of the puzzle and find one other clue in the same row or column. Write the factors for those two clues in the appropriate cells and then work your way down the puzzle cell by cell until you have written the numbers from 1 to 10 in both the first column and the top row.

Level 3 puzzles are a jump from level 2, but still significantly easier than any of the higher level puzzles, because the needed clues are kept in a logical order from the top of the puzzle to the bottom. You can print the puzzles above and work on them or you can access the whole collection in an excel sheet: Level 3’s from 2018. Print them or type the answers directly in excel.

Now I’ll share some information about the number 1345:

  • 1345 is a composite number.
  • Prime factorization: 1345 = 5 × 269
  • 1345 has no exponents greater than 1 in its prime factorization, so √1345 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 1345 has exactly 4 factors.
  • The factors of 1345 are outlined with their factor pair partners in the graphic below.

1345 is the sum of two squares in two different ways:
36² + 7² = 1345
33² + 16² = 1345

1345 is also the hypotenuse of FOUR Pythagorean triples:
345-1300-1345 which is 5 times (69-260-269)
504-1247-1345 calculated from 2(36)(7), 36² – 7², 36² + 7²
807-1076-1345 which is (3-4-5) times 269
833-1056-1345 calculated from 33² – 16², 2(33)(16), 33² + 16²

How Far Away Is 1344 from the Nearest Prime Number?

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The Distance 1344 is from the Nearest Prime Number:

1344 is 17 numbers away from the nearest prime number. 17-away is a new record for distance to the closest prime number!

1344 will hold that record until prime number 2179 claims it with being 18-away from the nearest prime:
2161–(18 composite numbers)–2179–(24 composite numbers)–2203

OEIS.org mentioned the previous two numbers, 1342 and 1343, breaking the 15-away and the 16 away records respectively, but didn’t mention this one. I guess it considered 1344 being the order of a perfect group to be more important.

Factors of 1344:

  • 1344 is a composite number.
  • Prime factorization: 1344 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 7, which can be written 1344 = 2⁶ × 3 × 7
  • 1344 has at least one exponent greater than 1 in its prime factorization so √1344 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1344 = (√64)(√21) = 8√21
  • The exponents in the prime factorization are 6, 1 and 1. Adding one to each exponent and multiplying we get (6 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 1344 has exactly 28 factors.
  • The factors of 1344 are outlined with their factor pairs in the graphic below.

Sum-Difference Puzzles:

84 has six factor pairs. One of those pairs adds up to 25, and another one subtracts to 25. Put the factors in the appropriate boxes in the first puzzle.

1344 has fourteen factor pairs. One of the factor pairs adds up to ­100, and a different one subtracts to 100. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

 

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