A Multiplication Based Logic Puzzle

Archive for the ‘Level 3 Puzzle’ Category

797 and Mathematical Ways to Love

Mathwithbaddrawings.com has some thoughtful and entertaining Ways to Tell a Mathematician that you love them.

Artful Maths wrote a post that includes beautiful mathematical origami valentines and a “string art” cardioid that is made with a pencil instead of string.

This puzzle could be another mathematical way to show some love:

797-love-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-795-799

Here are a few facts about the number 797:

797 is a palindrome in three bases:

  • 797 BASE 10 because 7(100) + 9(10) + 7(1) = 797
  • 565 BASE 12 because 5(144) + 6(12) + 5(1) = 797
  • 494 BASE 13 because 4(169) + 9(13) + 4(1) = 797

But there’s one more palindromic fact about the number 797: It is the sum of two square numbers that are also palindromes!

  • 797 = 121 + 676. Note that 11² = 121 and 26² = 676.

Since it is the sum of two squares, 797 will also be the hypotenuse a Pythagorean triple:

  • 555 – 572 – 797 calculated from 26² – 11², 2(26)(11), 26² + 11².

797 is the sum of three squares seven different ways:

  • 28² + 3² + 2² = 797
  • 27² + 8² + 2² = 797
  • 24² + 14² + 5² = 797
  • 24² + 11² + 10² = 797
  • 22² + 13² + 12² = 797
  • 21² + 16² + 10² = 797
  • 20² + 19² + 6² = 797

797 is also the sum of the 15 prime numbers from 23 to 83:

  • 23 + 29 + 31 + 37+ 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 797

Here is the most basic information about the number 797:

  • 797 is a prime number.
  • Prime factorization: 797 is prime and cannot be factored.
  • The exponent of prime number 797 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 797 has exactly 2 factors.
  • Factors of 797: 1, 797
  • Factor pairs: 797 = 1 x 797
  • 797 has no square factors that allow its square root to be simplified. √797 ≈ 28.231188.

How do we know that 797 is a prime number? If 797 were not a prime number, then it would be divisible by at least one prime number less than or equal to √797 ≈ 28.2. Since 797 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 797 is a prime number.

Here’s another way we know that 797 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 11² = 797 with 26 and 11 having no common prime factors, 797 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √797 ≈ 28.2. Since 797 is not divisible by 5, 13, or 17, we know that 797 is a prime number.

797-factor-pairs

 

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790 and Level 3

  • 790 is a composite number.
  • Prime factorization: 790 = 2 x 5 x 79
  • 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 x 2 x 2 = 8. Therefore 790 has exactly 8 factors.
  • Factors of 790: 1, 2, 5, 10, 79, 158, 395, 790
  • Factor pairs: 790 = 1 x 790, 2 x 395, 5 x 158, or 10 x 79
  • 790 has no square factors that allow its square root to be simplified. √790 ≈ 28.106939.

790-factor-pairs

Here is today’s puzzle:

790-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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Here’s a little more about the number 790:

There are four ways to make 790 using three squares:

  • 21² + 18² + 5² = 790
  • 27² + 6² + 5² = 790
  • 23² + 15² + 6² = 790
  • 22² + 15² + 9² = 790

790 is the hypotenuse of Pythagorean triple 474-632-790 which is 3-4-5 times 158.

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784 and Level 3

If you know the multiplication facts up to 12 x 12, then it is obvious that 784 is divisible by 7. However, 784 is divisible by a whole lot more numbers than that. 784 can actually be evenly divided by 15 different numbers. Hmm, 15, that’s an odd number. A number’s factors always come in pairs. One of its factor pairs must contain the same factor twice, meaning 784 is a perfect square!

  • 784 is a composite number.
  • Prime factorization: 784 = 2 x 2 x 2 x 2 x 7 x 7, which can be written 784 = (2^4) x (7^2)
  • The exponents in the prime factorization are 4 and 2. Adding one to each and multiplying we get (4 + 1)(2 + 1) = 5 x 3 = 15. Therefore 784 has exactly 15 factors.
  • Factors of 784: 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784
  • Factor pairs: 784 = 1 x 784, 2 x 392, 4 x 196, 7 x 112, 8 x 98, 14 x 56, 16 x 49, or 28 x 28
  • 784 is a perfect square. √784 = 28

784-factor-pairs

But that’s not the only thing remarkable about this perfect square: √784 is 28, the 7th triangular number, so like all other squared triangular numbers 784 has this additional property:

784-sum-of-consecutive-cubes

Just as 784 is a perfect square, five of the twelve clues in today’s puzzle are also perfect squares. But don’t let that fact trick you into writing the same factor in both the first column and the top row every time!

784-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s more about the number 784:

784 can be written as the sum of three squares in exactly one way:

  • 24² + 12² + 8² = 784

784 is also a palindrome in Bases 13, 17, and 27:

  • 484 BASE 13; note that 4(169) + 8(13) + 4(1) = 784
  • 2C2 BASE 17 (C is 12 base 10); note that 2(289) + 12(17) + 2(1) = 784
  • 121 BASE 27; note that 1(27²) + 2(27) + 1(1) = 784

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784-factors

 

 

776 and Level 3

  • 776 is a composite number.
  • Prime factorization: 776 = 2 x 2 x 2 x 97, which can be written 776 = (2^3) x 97
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 776 has exactly 8 factors.
  • Factors of 776: 1, 2, 4, 8, 97, 194, 388, 776
  • Factor pairs: 776 = 1 x 776, 2 x 388, 4 x 194, or 8 x 97
  • Taking the factor pair with the largest square number factor, we get √776 = (√4)(√194) = 2√194 ≈ 27.85677655

776-factor-pairs

Try solving today’s puzzle:

776 Puzzle

Print the puzzles or type the solution on this excel file: 10-factors-2016

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Here’s more about the number 776:

776 is the sum of two squares: 26² + 10² =776.

776 is the hypotenuse of Pythagorean triple 520-576-776 so 520² + 576² = 776².

776 is also the sum of three squares five different ways:

  • 26² + 8² + 6² = 776
  • 24² + 14² + 2² = 776
  • 24² + 10² + 10² = 776
  • 22² + 16² + 6² = 776
  • 18² + 16² + 14² = 776

776 is a palindrome in three other bases:

  • 646 BASE 11; note that 6(121) + 4(11) + 6(1) = 776
  • 272 BASE 18; note that 2(18²) + 7(18) + 2(1) = 776
  • 161 BASE 25; note that 1(25²) + 6(25) + 1(1) = 776

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776 Factors

770 and Level 3

  • 770 is a composite number.
  • Prime factorization: 770 = 2 x 5 x 7 x 11
  • 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 x 2 x 2 x 2 = 16. Therefore 770 has exactly 16 factors.
  • Factors of 770: 1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770
  • Factor pairs: 770 = 1 x 770, 2 x 385, 5 x 154, 7 x 110, 10 x 77, 11 x 70, 14 x 55, or 22 x 35
  • 770 has no square factors that allow its square root to be simplified. √770 ≈ 27.74887.

770-factor-pairs

Here is a puzzle for you to solve:

770 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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Here is more information about the number 770:

Because 5 is one of its factors, 770 is the hypotenuse of a Pythagorean triple, and 462² + 616² = 770².

770 is the sum of the squares of three consecutive numbers: 15² + 16² + 17² = 770.

770 can also be written as the sum of three squares seven other ways:

  • 27² + 5² + 4² = 770
  • 25² + 12² + 1² = 770
  • 25² + 9² + 8² = 770
  • 24² + 13² + 5² = 770
  • 23² + 15² + 4² = 770
  • 20² + 19² + 3² = 770
  • 20² + 17² + 9² = 770

770 is palindrome MM in Base 34 (M = 22 base 10); note that 22(34) + 22(1) = 770.

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770 Factors


763 and Level 3

It’s obvious that 763 is divisible by 7 so it is a composite number.

  • 763 is a composite number.
  • Prime factorization: 763 = 7 x 109
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 763 has exactly 4 factors.
  • Factors of 763: 1, 7, 109, 763
  • Factor pairs: 763 = 1 x 763 or 7 x 109
  • 763 has no square factors that allow its square root to be simplified. √763 ≈ 27.6224546.

763-factor-pairs

Now try solving today’s puzzle:

763 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s a little more about the number 763:

763 is the sum of consecutive numbers three different ways:

  • 381 + 382 = 763; that’s 2 consecutive numbers.
  • 106 + 107 + 108 + 109 + 110 + 111 + 112 = 763; that’s 7 consecutive numbers.
  • 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 = 763; that’s 14 consecutive numbers.

763 is also the sum of consecutive prime numbers two different ways:

  • 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 = 763; that’s 9 consecutive primes.
  • 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 763; that’s 17 consecutive primes.

763 is the hypotenuse of a Pythagorean triple, and 420² + 637² = 763².

763 is also the sum of three squares two different ways:

  • 27² + 5² + 3² = 763
  • 23² + 15² + 3² = 763

763^4 = 338,920,744,561, a number in which every digit appears at least one time. Stetson.edu informs us 763 is the smallest number whose 4th power can make that claim.

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763 Factors

756 and Level 3

  • 756 is a composite number.
  • Prime factorization: 756 = 2 x 2 x 3 x 3 x 3 x 7, which can be written 756 = (2^2) x (3^3) x 7
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 756 has exactly 24 factors.
  • Factors of 756: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756
  • Factor pairs: 756 = 1 x 756, 2 x 378, 3 x 252, 4 x 189, 6 x 126, 7 x 108, 9 x 84, 12 x 63, 14 x 54, 18 x 42, 21 x 36 or 27 x 28
  • Taking the factor pair with the largest square number factor, we get √756 = (√21)(√36) = 6√21 ≈ 27.495454.

756-factor-pairs

 

756 has many factors and, therefore, it has many possible factor trees. Here are three of them:

756 factor trees

Here’s a level 3 Find the Factors puzzle for you to solve, too:

756 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s a few more thoughts on the number 756:

The last two digits of 756 is divisible by 4 so 756 is divisible by 4.

756 is formed from 3 consecutive numbers (5, 6, 7) so it is divisible by 3. The middle number is divisible by 3 so 756 is also divisible by 9.

756 can be written as the sum of consecutive numbers seven ways:

  • 251 + 252 + 253 = 756; that’s 3 consecutive numbers.
  • 105 + 106 + 107 + 108 + 109 + 110 + 111 = 756; that’s 7 consecutive numbers.
  • 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 = 756; that’s 8 consecutive numbers.
  • 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 756; that’s 9 consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 756; that’s 21 consecutive numbers.
  • 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 = 756; that’s 24 consecutive numbers.
  • 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  = 756; that’s 27 consecutive numbers.

756 is also the sum of six consecutive prime numbers: 109 + 113 + 127+ 131 + 137+ 139 = 756.

756 can be written as the sum of three squares four different ways. (Notice that all of the squares are even):

  • 26² + 8² + 4² = 756
  • 24² + 12² + 6² = 756
  • 22² + 16² + 4² = 756
  • 20² + 16² + 10² = 756

756 is a palindrome in two other bases:

  • 11011 BASE 5; note that 1(625) + 1(125) + 0(25) + 1(5) + 1(1) = 756.
  • LL BASE 35 (L is 21 base 10); note that 21(35) + 21(1) = 756.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

756 Factors

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