Month: July 2019

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)

 

 

1402 Mystery Level

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Mystery level puzzles may be very difficult or relatively easy. How much trouble will this one be? You’ll have to try it to see!

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

Now I’ll tell you a little bit about the number 1402:

  • 1402 is a composite number.
  • Prime factorization: 1402 = 2 × 701.
  • 1402 has no exponents greater than 1 in its prime factorization, so √1402 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 1402 has exactly 4 factors.
  • The factors of 1402 are outlined with their factor pair partners in the graphic below.

1402 is the hypotenuse of a Pythagorean triple:
31² + 21² = 1402

1402 is the hypotenuse of a Pythagorean triple:
520-1302-1402 which is 2 times (260-651-701)
and can also be calculated from 2(31)(21), 31² – 21², 31² + 21²

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?

Prime Factorization of the Hundred Numbers up to 1400

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Almost one-third of the numbers from 1301 to 1400 have 4 factors. Only 1/5 of the numbers have 8 factors.

Since 1/3 is significantly bigger than 1/5, the amount of factors for these numbers wouldn’t make a very exciting horse race. Here is the breakdown:

  • 11 numbers had 2 factors
  • 1 number had 3 factors
  • 32 numbers had 4 factors
  • 7 numbers had 6 factors
  • 20 numbers had 8 factors
  • 2 numbers had 10 factors
  • 13 numbers had 12 factors
  • 4 numbers had 16 factors
  • 1 number had 18 factors
  • 2 numbers had 20 factors
  • 5 numbers had 24 factors
  • 1 number had 28 factors
  • 1 number had 32 factors

The rosy looking numbers have square roots that can be simplified, and that is only 37% of the numbers listed.

You may not expect it, but 1400 is one of the numbers with 24 factors. Let me tell you a little bit about 1400 and why it has so many factors:

  • 1400 is a composite number.
  • Prime factorization: 1400 = 2 × 2 × 2 × 5 × 5 × 7, which can be written 1400 = 2³ × 5² × 7
  • 1400 has at least one exponent greater than 1 in its prime factorization so √1400 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1400 = (√100)(√14) = 10√14
  • The exponents in the prime factorization are 3, 2, and 1. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1400 has exactly 24 factors.
  • The factors of 1400 are outlined with their factor pair partners in the graphic below.

1400 is the hypotenuse of TWO Pythagorean triples:
392-1344-1400 which is (7-24-25) times 56
840-1120-1400 which is (3-4-5) times 280

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