Puzzles

1378 and Level 3

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The first row with a clue has a 14 in it. Use that 14 and the clue in the same column, to figure out where to put one of the factor pairs of 14 in this puzzle.  Only use factor pairs where both numbers are from 1 to 10. Then work your way down the puzzle, row by row until you have found all the factors of this level 3 puzzle. The completed puzzle will look like a multiplication table with the factors in a different order.

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here is some information about the puzzle number, 1378:

  • 1378 is a composite number.
  • Prime factorization: 1378 = 2 × 13 × 53
  • 1378 has no exponents greater than 1 in its prime factorization, so √1378 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 1378 has exactly 8 factors.
  • The factors of 1378 are outlined with their factor pair partners in the graphic below.

1378 is the 52nd triangular number because (52)(53)/2 = 1378.

1378 is the hypotenuse of FOUR Pythagorean triples:
222-1360-1378 which is 2 times (111-680-689)
530-1272-1378 which is (5-12-13) times 106
728-1170-1378 which is 26 times (28-45-53)
800-1122-1378 which is 2 times (400-561-689)

1377 Easter Basket Challenge

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Occasionally,  we hear that the number of Easter eggs that are found is one or two less than the number of eggs that were hidden. Still most of the time, all the eggs and candies do get found. You really have no trouble finding all those goodies, and the Easter Egg Hunt seems like it is over in seconds.  You can find Easter Eggs but can you find factors? Here’s an Easter Basket Find the Factors 1 – 10 Challenge Puzzle for you. I guarantee it won’t be done in seconds. Can you find all the factors? I dare you to try!

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll mention a few facts about the number 1377:

  • 1377 is a composite number.
  • Prime factorization: 1377 = 3 × 3 × 3 × 3 × 17, which can be written 1377 = 3⁴ × 17
  • 1377 has at least one exponent greater than 1 in its prime factorization so √1377 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1377 = (√81)(√17) = 9√17
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1377 has exactly 10 factors.
  • The factors of 1377 are outlined with their factor pair partners in the graphic below.

1377 is the sum of two squares:
36² + 9² = 1377

1377 is the hypotenuse of a Pythagorean triple:
648-1215-1377 which is (8-15-17) times 81
and can also be calculated from 2(36)(9), 36² – 9², 36² + 9²

1374 and Level 2

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Can you find the factors that will turn this puzzle into a multiplication table whose products are simply not in the usual order?

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are a few facts about the puzzle number, 1374:

  • 1374 is a composite number.
  • Prime factorization: 1374 = 2 × 3 × 229
  • 1374 has no exponents greater than 1 in its prime factorization, so √1374 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 1374 has exactly 8 factors.
  • The factors of 1374 are outlined with their factor pair partners in the graphic below.

1374 is the hypotenuse of a Pythagorean triple:
360-1326-1374 which is 6 times (60-221-229)

1373 and Level 1

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This puzzle is really just a multiplication table but the factors are missing and the products are not in order. You can figure out where the factors go, and then the clues will all make sense.


Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some facts about the puzzle number, 1373:

  • 1373 is a prime number.
  • Prime factorization: 1373 is prime.
  • 1373 has no exponents greater than 1 in its prime factorization, so √1373 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1373 has exactly 2 factors.
  • The factors of 1373 are outlined with their factor pair partners in the graphic below.

How do we know that 1373 is a prime number? If 1373 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1373. Since 1373 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1373 is a prime number.

1373 is the sum of two squares:
37² + 2² = 1373

1373 is the hypotenuse of a Pythagorean triple:
148-1365-1373 calculated from 2(37)(2), 37² – 2², 37² + 2²

Here’s another way we know that 1373 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 37² + 2² = 1373 with 37 and 2 having no common prime factors, 1373 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1373. Since 1373 is not divisible by 5, 13, 17,  29, or 37, we know that 1373 is a prime number.

1365 Shamrock Mystery

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Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story.  Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you!

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Now I’ll share some information about the number 1365:

  • 1365 is a composite number.
  • Prime factorization: 1365 = 3 × 5 × 7 × 13
  • 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 × 2 × 2 × 2 = 16. Therefore 1365 has exactly 16 factors.
  • Factors of 1365: 1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365
  • Factor pairs: 1365 = 1 × 1365, 3 × 455, 5 × 273, 7 × 195, 13 × 105, 15 × 91, 21 × 65, or 35 × 39
  • 1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591

1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273

1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16

I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help:

1363 and Level 6

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The common factors of 60 and 30 allowed in the puzzle are 5, 6, and 10. Which one is the logical choice? Look at the other clues in the puzzle and you should be able to eliminate two of the choices.

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few thoughts about the puzzle number, 1363:

  • 1363 is a composite number.
  • Prime factorization: 1363 = 29 × 47
  • 1363 has no exponents greater than 1 in its prime factorization, so √1363 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 1363 has exactly 4 factors.
  • The factors of 1363 are outlined with their factor pair partners in the graphic below.

1363 is the hypotenuse of a Pythagorean triple:
940-987-1363 which is (20-21-29) times 47

1361 and Level 5

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If you carefully use logic instead of guessing and checking you can find the unique solution to this puzzle without tearing your hair out!

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1361:

  • 1361 is a prime number.
  • Prime factorization: 1361 is prime.
  • 1361 has no exponents greater than 1 in its prime factorization, so √1361 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1361 has exactly 2 factors.
  • The factors of 1361 are outlined with their factor pair partners in the graphic below.

How do we know that 1361 is a prime number? If 1361 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1361. Since 1361 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1361 is a prime number.

1361 is the first prime number after 1327. That was 34 numbers ago!

1361 is the sum of two squares:
31² + 20² = 1361

1361 is the hypotenuse of a Pythagorean triple:
561-1240-1361 calculated from 31² – 20², 2(31)(20), 31² + 20²

Here’s another way we know that 1361 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 20² = 1361 with 31 and 20 having no common prime factors, 1361 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1361. Since 1361 is not divisible by 5, 13, 17, or 29, we know that 1361 is a prime number.

1360 and Level 4

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There is only one way to write the numbers 1 to 12 in the top row and the first column of this puzzle so that the puzzle could turn into a multiplication table. Can you find the solution?

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1360:

  • 1360 is a composite number.
  • Prime factorization: 1360 = 2 × 2 × 2 × 2 × 5 × 17, which can be written 1360 = 2⁴ × 5 × 17
  • 1360 has at least one exponent greater than 1 in its prime factorization so √1360 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1360 = (√16)(√85) = 4√85
  • 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 1360 has exactly 20 factors.
  • The factors of 1360 are outlined with their factor pair partners in the graphic below.

1360 is the sum of two squares in two different ways:
36² + 8² = 1360
28² + 24² = 1360

1360 is the hypotenuse of FOUR Pythagorean triples:
208-1344-1360 which is 16 times (13-84-85)
576-1232-1360 which is 16 times (36-77-85)
640-1200-1360 which is (8-15-17) times 80
816-1088-1360 which is (3-4-5) times 272

1359 and Level 3

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The first row with a clue has a 6 in it. Use that 6 and another clue that goes with it, to figure out where to put the factors of 6 in this puzzle. Then work your way down the puzzle, row by row until you have found all the factors of this level 3 puzzle.

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few facts about the number 1359:

  • 1359 is a composite number.
  • Prime factorization: 1359 = 3 × 3 × 151, which can be written 1359 = 3² × 151
  • 1359 has at least one exponent greater than 1 in its prime factorization so √1359 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1359 = (√9)(√151) = 3√151
  • 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 1359 has exactly 6 factors.
  • The factors of 1359 are outlined with their factor pair partners in the graphic below.

1359 is the difference of two squares in three different ways:
680² – 679² = 1359
228² – 225² = 1359
80² – 71² = 1359

1358 and Level 2

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If you are familiar with a basic 12 × 12 multiplication table, then you can solve this puzzle. The clues aren’t in the same order as they are in the table, but that only makes it a little more challenging.

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here’s a little bit about the number 1358:

  • 1358 is a composite number.
  • Prime factorization: 1358 = 2 × 7 × 97
  • 1358 has no exponents greater than 1 in its prime factorization, so √1358 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 1358 has exactly 8 factors.
  • The factors of 1358 are outlined with their factor pair partners in the graphic below.

910-1008-1358 which is 14 times (65-72-97)

OEIS.org informs us that 1358!!!! + 1 is a prime number.