A Multiplication Based Logic Puzzle

Posts tagged ‘mathematics’

837 and Level 3

837 is repdigit RR in BASE 30 (R is 27 in base 10). All that means is that 27(30¹) + 27(30º) = 837, which naturally follows from the fact that 27(30 + 1) = 837.

837 has four odd factor pairs, so 837 can be written as the difference of two squares four different ways:

  • 837 × 1 = 837 means 419² – 418² = 837
  • 279 × 3 = 837 means 141² – 138² = 837
  • 93 × 9 = 837 means 51² – 42² = 837
  • 31 × 27 = 837 means 29² – 2² = 837

Hmm…837 is only four numbers away from the next perfect square, 841.

Print the puzzles or type the solution on this excel file: 10-factors-835-842

  • 837 is a composite number.
  • Prime factorization: 837 = 3 × 3 × 3 × 31, which can be written 837 = 3³ × 31
  • 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 837 has exactly 8 factors.
  • Factors of 837: 1, 3, 9, 27, 31, 93, 279, 837
  • Factor pairs: 837 = 1 × 837, 3 × 279, 9 × 93, or 27 × 31
  • Taking the factor pair with the largest square number factor, we get √837 = (√9)(√93) = 3√93 ≈ 28.93095

 

 

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834 and Level 6

834 is the sum of consecutive prime numbers two different ways:

  • 127 + 131 + 137 + 139 + 149 + 151 = 834; that’s six consecutive primes
  • 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 834; that’s fourteen consecutive primes

The ONLY Pythagorean triple that contains the number 834 is 834 – 173888 – 173890.

  • 834 is a composite number.
  • Prime factorization: 834 = 2 × 3 × 139
  • 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 834 has exactly 8 factors.
  • Factors of 834: 1, 2, 3, 6, 139, 278, 417, 834
  • Factor pairs: 834 = 1 × 834, 2 × 417, 3 × 278, or 6 × 139
  • 834 has no square factors that allow its square root to be simplified. √834 ≈ 28.879058

There was a solar eclipse in the United States today. People where I lived were able to experience 91.32% obstruction of the sun. I love this interactive map of today’s eclipse and past and future ones as well.

Several people have taken and shared marvelous pictures of the eclipse.

Here are a few tweets I saw about eclipses on twitter:

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833 and Level 5

Print the puzzles or type the solution on this excel file: 12 factors 829-834

Wrap Your Mind Around This:  × 17 = 833 = 28² + 7²

833 is the hypotenuse of a Pythagorean triple:

  • 392-735-833 calculated from 2(28)(7), 28² – 7², 28² + 7²

833 can be written as the sum of consecutive prime numbers two ways. One of those ways starts with one of its prime factors, 17:

  • 833 is the sum of the seventeen prime numbers from 17 to 83
  • 833 is also the sum of the eleven prime numbers from 53 to 101

Since 833 has three factor pairs where both factors are odd, it can be written as the difference of two squares three different ways:

  • 833 × 1 = 833 means 417² – 416² = 833
  • 119 × 7 = 833 means 63² – 56² = 833
  • 49 × 17 = 833 means 33² – 16² = 833

The 41st triangular number will be 861. We must use less than 41 consecutive numbers if we want to express 833 as the sum of consecutive numbers. 833 has 3 odd factors (1, 7, 17) less than 41. Thus 833 can be written as the sum of 7 consecutive numbers and as the sum of 17 consecutive numbers. Notice 833’s factor pairs below highlighted in red.

  • 833 = 116 + 117 + 118 + 119 + 120 + 121 + 122; that’s 7 consecutive numbers
  • 833 = 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57; that’s 17 consecutive numbers

The factor of 833 that is the highest power of 2 is 1 because 2º=1. Each of those odd factors, (1, 7, 17), times 2 × 1 is still less than 40, so 833 can also be written as the sum of 2 consecutive numbers, the sum of 14 consecutive numbers, and the sum of 34 consecutive numbers:

  • 833 = 416 + 417; that’s 2 consecutive numbers
  • 833 = 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66; that’s 14 consecutive numbers
  • 833 = 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41; that’s 34 consecutive numbers

Here is 833’s factoring information:

  • 833 is a composite number.
  • Prime factorization: 833 = 7 × 7 × 17, which can be written 833 = 7² × 17
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 833 has exactly 6 factors.
  • Factors of 833: 1, 7, 17, 49, 119, 833
  • Factor pairs: 833 = 1 × 833, 7 × 119, or 17 × 49
  • Taking the factor pair with the largest square number factor, we get √833 = (√49)(√17) = 7√17 ≈ 28.861739

832 and Level 4

Print the puzzles or type the solution on this excel file: 12 factors 829-834

832 has many factors, but it can be written as the sum of consecutive numbers only one way:

  • 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 = 832; that’s thirteen consecutive numbers.

832 can be written as the difference of two squares five different ways because it has five factor pairs in which both numbers are even:

  • 416 × 2 = 832 means 209² – 207² = 832
  • 208 × 4 = 832 means 106² – 102² = 832
  • 104 × 8 = 832 means 56² – 48² = 832
  • 52 × 16 = 832 means 34² – 18² = 832
  • 32 × 26 = 832 means 29² – 3² = 832

832 is also the sum of two squares:

  • 24² + 16² = 832

832 is the hypotenuse of a Pythagorean triple:

  • 320-768-832 calculated from 24² – 16², 2(24)(16), 24² + 16²
  • 320-768-832 is also 64 times (5-12-13)

832 is repdigit QQ in BASE 31 (Q is 26 base 10). That’s because 26(31) + 26(1) = 832, which is the same as saying 26 × 32 = 832.

  • 832 is a composite number.
  • Prime factorization: 832 = 2 × 2 × 2 × 2 × 2 × 2 × 7, which can be written 832 = 2⁶ × 7
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 832 has exactly 14 factors.
  • Factors of 832: 1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 208, 416, 832
  • Factor pairs: 832 = 1 × 832, 2 × 416, 4 × 208, 8 × 104, 13 × 64, 16 × 52, or 26 × 32
  • Taking the factor pair with the largest square number factor, we get √832 = (√64)(√13) = 8√13 ≈ 28.8444102

700 Pick Your Pony! Who will win this Amount of Factors Horse Race?

  • 700 is a composite number.
  • Prime factorization: 700 = 2 x 2 x 5 x 5 x 7, which can be written 700 = (2^2) x (5^2) x 7
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 700 has exactly 18 factors.
  • Factors of 700: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700
  • Factor pairs: 700 = 1 x 700, 2 x 350, 4 x 175, 5 x 140, 7 x 100, 10 x 70, 14 x 50, 20 x 35 or 25 x 28
  • Taking the factor pair with the largest square number factor, we get √700 = (√100)(√7) = 10√7 ≈ 26.457513.

Because this is my 700th post, I think I’ll have another horse race. Some numbers from 601 to 700 have exactly 2 factors, 4 factors, and so forth up to 24 factors. (Only perfect squares can have an odd number of factors.)

Which number from 1 to 24 will win this amount of factors horse race? Which number will come in second place, or third place? Cheering for more than one pony will make the race even more interesting.

Here we see that the numbers 2, 6, & 8 are the first ones out of the gate. Click on the graphic to see the rest of this very thrilling horse race:

605

Every hundred posts I also like to focus on the percentage of numbers whose square roots can be simplified.

700 is divisible by 100 so its square root can easily be simplified: √700 = 10√7.

273 of the first 700 numbers have reducible square roots. That’s exactly 39%.

The rest of the numbers, 427, which is 41% of the first 700 numbers, do not have reducible square roots.

Here’s a table breaking down the amount of factors in each group of one hundred integers and the number of reducible square roots.

1-700 Amount of Factors

Here are some facts about the number 700.

700 is a palindrome in several bases:

  • 4A4 BASE 12; note A is equivalent to 1o in base 10, and 4(144) + 10(12) + 4(1) = 700
  • PP BASE 27; note P is equivalent to 25 in base 10, and 25(27) + 25(1) = 700
  • KK Base 34; note K is equivalent to 20 in base 10, and 20(34) + 20(1) = 700

700 is the sum of four consecutive prime numbers: 167 + 173 + 179 + 181.

Here is a beautiful painting of a horse race that I saw on twitter:

508 A Mathematical and Biblical Truth: 2 > 1

Two Is Better Than One Because...

Two are better than one; because they have a good reward for their labour.

For if they fall, the one will lift up his fellow: but woe to him that is alone when he falleth; for he hath not another to help him up.

Ecclesiastes 4: 9 – 10

—————————————————————————————————

508 is the sum of some consecutive prime numbers, and at least one of those prime numbers is in its prime factorization. Can you figure out what those consecutive primes are?

  • 508 is a composite number.
  • Prime factorization: 508 = 2 x 2 x 127, which can be written 508 = (2^2) x 127
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 508 has exactly 6 factors.
  • Factors of 508: 1, 2, 4, 127, 254, 508
  • Factor pairs: 508 = 1 x 508, 2 x 254, or 4 x 127
  • Taking the factor pair with the largest square number factor, we get √508 = (√4)(√127) = 2√127 ≈ 22.53 88 55 33 9

336 and Level 2

  • 336 is a composite number.
  • Prime factorization: 336 = 2 x 2 x 2 x 2 x 3 x 7, which can be written 336 = (2^4) x 3 x 7
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 336 has exactly 20 factors.
  • Factors of 336: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336
  • Factor pairs: 336 = 1 x 336, 2 x 168, 3 x 112, 4 x 84, 6 x 56, 7 x 48, 8 x 42, 12 x 28, 14 x 24 or 16 x 21
  • Taking the factor pair with the largest square number factor, we get √336 = (√16)(√21) = 4√21 ≈ 18.330

The sixteen clues given in this puzzle are all you need to complete this multiplication table!

2014-52 Level 2

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 2 Factors

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