Find the Factors Challenge

1467 Challenge Puzzle

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Challenge Puzzle:

Use logic to solve this Challenge Puzzle. The given clues work together to make finding the unique solution a little easier than usual. Have fun with it!

Print the puzzles or type the solution in this excel file: 10 Factors 1454-1467

Factors of 1467:

1467 is divisible by 3 because 1 + 4 = 5 and 5, 6, 7 are three consecutive numbers. Since the middle number, 6, is divisible by 3, we know that 1467 is also divisible by 9.

Of course, you could also add up the digits of 1467 to get 1 + 4 + 6 + 7 = 27, a number divisible by both 3 and 9, to know that 1467 is divisible by both 3 and 9.

  • 1467 is a composite number.
  • Prime factorization: 1467 = 3 × 3 × 163, which can be written 1467 = 3² × 163
  • 1467 has at least one exponent greater than 1 in its prime factorization so √1467 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1467 = (√9)(√163) = 3√163
  • 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 1467 has exactly 6 factors.
  • The factors of 1467 are outlined with their factor pair partners in the graphic below.

 

 

1429 Find the Factors Challenge

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I wanted today’s puzzle to look like a big candy bar, but I don’t think I succeeded. I hope you will still think it is the best treat you got today! Good luck!

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you some facts about the number 1429:

  • 1429 is a prime number.
  • Prime factorization: 1429 is prime.
  • 1429 has no exponents greater than 1 in its prime factorization, so √1429 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1429 has exactly 2 factors.
  • The factors of 1429 are outlined with their factor pair partners in the graphic below.

How do we know that 1429 is a prime number? If 1429 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1429. Since 1429 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1429 is a prime number.

OEIS.org informs us that 1429² = 2,042,041. That’s the smallest perfect square whose first three digits are repeated in order by the next three digits.

1429 is the sum of two squares:
30² + 23² = 1429

1429 is the hypotenuse of a primitive Pythagorean triple:
371-1380-1429 calculated from 30² – 23², 2(30)(23), 30² + 23²

Here’s another way we know that 1429 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 23² = 1429 with 30 and 23 having no common prime factors, 1429 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1429. Since 1429 is not divisible by 5, 13, 17, 29, or 37, we know that 1429 is a prime number.

 

1418 Challenge Puzzle

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The 19 clues in this Find the Factors Challenge Puzzle are enough to find its unique solution. Can you find it?

Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll write a few facts about the puzzle number, 1418:

  • 1418 is a composite number.
  • Prime factorization: 1418 = 2 × 709.
  • 1418 has no exponents greater than 1 in its prime factorization, so √1418 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 1418 has exactly 4 factors.
  • The factors of 1418 are outlined with their factor pair partners in the graphic below.

1418 is the sum of two squares:
37² + 7² = 1418

518-1320-1418 calculated from 2(37)(7), 37² – 7², 37² + 7².
It is also 2 times (259-660-709)

 

1403 Multiplication Table Challenge

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Just because you’re not in elementary school anymore doesn’t mean that the multiplication table can’t be a challenge. This one certainly is. Can you write the numbers 1 to 10 in the four factor areas so that this multiplication table works with the given clues? Don’t get discouraged; it will probably take you at least 15 minutes just to put those factors in the right places.

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

Now I’ll share some information about the puzzle number, 1403:

  • 1403 is a composite number.
  • Prime factorization: 1403 = 23 × 61.
  • 1403 has no exponents greater than 1 in its prime factorization, so √1403 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 1403 has exactly 4 factors.
  • The factors of 1403 are outlined with their factor pair partners in the graphic below.

1403 is the hypotenuse of a Pythagorean triple:
253-1380-1403 which is 23 times (11-60-61)

 

 

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

1350 Logic is at the Heart of This Puzzle

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Today’s Puzzle:

By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!

If you need some tips on how to get started on this puzzle, check out this video:

Factors of 1350:

  • 1350 is a composite number.
  • Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
  • The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
  • Factors of 1350: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350
  • Factor pairs: 1350 = 1 × 1350, 2 × 675, 3 × 450, 5 × 270, 6 × 225, 9 × 150, 10 × 135, 15 × 90, 18 × 75, 25 × 54, 27 × 50 or 30 × 45
  • Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

1350 has twelve factor pairs. One of the factor pairs adds up to ­75, and a different one subtracts to 75. 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?

More about the Number 1350:

1350 is the sum of consecutive prime numbers two ways:
It is the sum of the fourteen prime numbers from 67 to 131, and
673 + 677 = 1350

1350 is the hypotenuse of two Pythagorean triples:
810-1080-1350 which is (3-4-5) times 270
378-1296-1350 which is (7-24-25) times 54

1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350

1349 A Rejected Puzzle

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I was in the mood to make a Find the Factors Challenge Puzzle that used the numbers from 1 to 12 as the factors. I’ve never made such a large puzzle before, but after I made it, I rejected it. All the puzzles I make must meet certain standards: they must have a unique solution, and that solution must be obtainable by using logic. Although the “puzzle” below has a unique solution, and you can fill in a few of the cells using logic, you would have to use guess and check to finish it. Besides that, you wouldn’t be able to know if you guessed right until almost the entire puzzle was completed. Thus, it doesn’t meet my standards.

Even though the puzzle was rejected, there were still some things about it that I really liked. In my next post, I’ll publish a slightly different puzzle that uses some of the same necessary logic that I appreciated but doesn’t rely on guess and check at all. This is NOT the first time I have tweaked a puzzle that didn’t initially meet my standards to make it acceptable. I just thought I would share the process this time. If you try to solve it, you will be able to see the problem with the puzzle yourself.

Now I’ll share some information about the number 1349:

  • 1349 is a composite number.
  • Prime factorization: 1349 = 19 × 71
  • 1349 has no exponents greater than 1 in its prime factorization, so √1349 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 1349 has exactly 4 factors.
  • The factors of 1349 are outlined with their factor pair partners in the graphic below.

1349 is the sum of 13 consecutive primes, and it is also the sum of three consecutive primes:
73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1349
443 + 449 + 457 = 1349

1319 Challenge Puzzle

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There are nineteen clues in this challenge puzzle, including two 50’s, two 20’s, two 8’s, and two 12’s. Some of those duplicates might make it more difficult for you to find the one and only solution to the puzzle. I’m very curious about how you do with it!

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

Now I’ll share some facts about the number 1319:

  • 1319 is a prime number.
  • Prime factorization: 1319 is prime.
  • The exponent of prime number 1319 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1319 has exactly 2 factors.
  • Factors of 1319: 1, 1319
  • Factor pairs: 1319 = 1 × 1319
  • 1319 has no square factors that allow its square root to be simplified. √1319 ≈ 36.31804

How do we know that 1319 is a prime number? If 1319 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1319 ≈ 36.3. Since 1319 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1319 is a prime number.

1310 Happy Birthday to My Brother, Andy

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Today is my brother’s birthday. He likes puzzles so I’ve made him a tough, challenging one. Still, he’ll probably figure it out in no time. Happy birthday, Andy!

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Like always, I’ll write what I’ve learned about a number. This time it’s 1310’s turn.

  • 1310 is a composite number.
  • Prime factorization: 1310 = 2 × 5 × 131
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1310 has exactly 8 factors.
  • Factors of 1310: 1, 2, 5, 10, 131, 262, 655, 1310
  • Factor pairs: 1310 = 1 × 1310, 2 × 655, 5 × 262, or 10 × 131
  • 1310 has no square factors that allow its square root to be simplified. √1310 ≈ 36.19392

1310 is the hypotenuse of a Pythagorean triple:
786-1048-1310 which is (3-4-5) times 262

As shown in their factor trees below, 1308, 1309, 1310, and 1311 each have three distinct prime numbers in their prime factorizations. They are the smallest set of four consecutive numbers with the same number of prime factors. 1309, 1310, and 1311 are also the smallest three consecutive numbers that have exactly the same number of factors and factor pairs. Thank you OEIS.org for alerting me to those facts.

 

1220 Challenge Puzzle

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The last challenge puzzle was particularly difficult. This one won’t be nearly as bad. Try it and see if you can figure it out!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Here are a few facts about the number 1220:

  • 1220 is a composite number.
  • Prime factorization: 1220 = 2 × 2 × 5 × 61, which can be written 1220 = 2² × 5 × 61
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1220 has exactly 12 factors.
  • Factors of 1220: 1, 2, 4, 5, 10, 20, 61, 122, 244, 305, 610, 1220
  • Factor pairs: 1220 = 1 × 1220, 2 × 610, 4 × 305, 5 × 244, 10 × 122, or 20 × 61
  • Taking the factor pair with the largest square number factor, we get √1220 = (√4)(√305) = 2√305 ≈ 34.9285

1220 is the sum of consecutive prime numbers: 607 + 613 = 1220

1220 is the sum of two squares two different ways:
32² + 14² = 1220
34² + 8² = 1220

1220 is the hypotenuse of four Pythagorean triples:
220-1200-1220 which is 20 times (11-60-61)
544-1092-1220 calculated from 2(34)( 8), 34² – 8², 34² + 8²
and is also 4 times (136-273-305)
828-896-1220 calculated from 32² – 14², 2(32)(14), 32² + 14²
and is also 4 times (207-224-305)
732-976-1220 which is (3-4-5) times 244