A Multiplication Based Logic Puzzle

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:

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.

761 and Level 1

• 761 is a prime number.
• Prime factorization: 761 is prime and cannot be factored.
• The exponent of prime number 761 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 761 has exactly 2 factors.
• Factors of 761: 1, 761
• Factor pairs: 761 = 1 x 761
• 761 has no square factors that allow its square root to be simplified. √761 ≈ 27.586228.

How do we know that 761 is a prime number? If 761 were not a prime number, then it would be divisible by at least one prime number less than or equal to √761 ≈ 27.6. Since 761 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 761 is a prime number.

There will be a lot about squares in today’s post, and all of the clues in today’s puzzle are perfect squares, too:

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

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Here’s more about the number 761:

Because 761 is a prime number, there is only one way to write it as the sum of consecutive numbers: 380 + 381 = 761.

However, because it is a centered square number, 761 can be written as the sum of consecutive square numbers: 20² + 19² = 761.

And because it is the sum of two consecutive square numbers, it is also the hypotenuse of a primitive Pythagorean triple: 39-760-761 which was calculated using 20² – 19², 2(20)(19), 20² + 19².

Notice also that 20 + 19 = 39, ⌊39²/2⌋ = 760, and ⌈39²/2⌉ = 761. (Those peculiar parenthesis mean⌊round down to the closest whole number⌋ and ⌈round up to the closest whole number⌉.)

In the 39 x 39 graphic shown above there are 760 white squares and 761 squares with some color. Thus, looking at that graphic is one way to look at the primitive Pythagorean triple 39-760-761.

All of that gives us another way to know that 761 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 20² + 19² = 761 with 20 and 19 having no common prime factors, 761 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √761 ≈ 27.6. Since 761 is not divisible by 5, 13, or 17, we know that 761 is a prime number.

761 can also be expressed as the sum of three squares TEN ways:

• 27² + 4² + 4² = 761
• 26² + 9² + 2² = 761
• 26² + 7² + 6² = 761
• 25² + 10² + 6² = 761
• 24² + 13² + 4² = 761
• 24² + 11² + 8² = 761
• 23² + 14² + 6² = 761
• 22² + 14² + 9² = 761
• 21² + 16² + 8² = 761
• 19² + 16² + 12² = 761

761 is a palindrome in two different bases:

• 535 BASE 12; note that 5(144) + 3(12) + 5(1) = 761.
• 1I1 BASE 20 (I = 18 base 10); note that 1(400) + 18(20) + 1(1) = 761.

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757 and Level 4

• 757 is a prime number.
• Prime factorization: 757 is prime and cannot be factored.
• The exponent of prime number 757 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 757 has exactly 2 factors.
• Factors of 757: 1, 757
• Factor pairs: 757 = 1 x 757
• 757 has no square factors that allow its square root to be simplified. √757 ≈ 27.51363.

How do we know that 757 is a prime number? If 757 were not a prime number, then it would be divisible by at least one prime number less than or equal to √757 ≈ 27.5. Since 757 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 757 is a prime number.

Here’s today’s puzzle. A logical way to solve it can be found in a table at the end of the post.

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

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Here’s another way we know that 757 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 9² = 757 with 26 and 9 having no common prime factors, 757 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √757 ≈ 27.5. Since 757 is not divisible by 5, 13, or 17, we know that 757 is a prime number.

757 is prime so the only way it can be written as the sum of consecutive numbers is 378 + 379 = 757.

757 is the sum of seven consecutive prime numbers: 97 + 101 + 103 + 107 + 109 + 113 + 127 = 757.

From Stetson.edu we learn this distinguishing fact about the number 757:

Here are some square facts about the number 757:

26² + 9² = 757, and 468² + 595² = 757². That primitive Pythagorean triple, 468-595-757, can be calculated from 2(26)(9), 26² – 9², 26² + 9².

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

• 24² + 10² + 9² = 757
• 18² + 17² + 12² = 757

Finally 757 is a palindrome in four different bases:

• 1001001 BASE 3; note that 1(3^6) + 0(3^5) + 0(3^4) + 1(3^3) + 0(3^2) + 0(3^1) + 1(3^0) = 757.
• 757 BASE 10; note that 7(100) + 5(10) + 7(1) = 757.
• 1F1 BASE 21 (F is 15 base 10); note that 1(21²) + 15(21) + 1(1) = 757.
• 111 BASE 27; note that 1(27²) + 1(27) + 1(1) = 757.

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743 and Level 5

• 743 is a prime number.
• Prime factorization: 743 is prime and cannot be factored.
• The exponent of prime number 743 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 743 has exactly 2 factors.
• Factors of 743: 1, 743
• Factor pairs: 743 = 1 x 743
• 743 has no square factors that allow its square root to be simplified. √743 ≈ 27.258026.

How do we know that 743 is a prime number? If 743 were not a prime number, then it would be divisible by at least one prime number less than or equal to √743 ≈ 27.3. Since 743 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 743 is a prime number.

Here is today’s puzzle. It can be solved using logic as explained in the table at the end of the post.

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

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Here is just a little more about the number 743:

743 is prime so it can be written as the sum of consecutive numbers only one way: 371 + 372 = 743.

743 is a palindrome in two bases:

• 616 BASE 11; note that 6(11²) + 1(11) + 6(1) = 743.
• 212 BASE 19; note that 2(19²) + 1(19) + 2(1) = 743.

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701 Some Virgács left by Mikulás

• 701 is a prime number.
• Prime factorization: 701 is prime and cannot be factored.
• The exponent of prime number 701 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 701 has exactly 2 factors.
• Factors of 701: 1, 701
• Factor pairs: 701 = 1 x 701
• 701 has no square factors that allow its square root to be simplified. √701 ≈ 26.4764046.

How do we know that 701 is a prime number? If 701 were not a prime number, then it would be divisible by at least one prime number less than or equal to √701 ≈ 26.4. Since 701 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 701 is a prime number.

Tonight Mikulás will visit the homes of children who sleep in Hungary. If they have been good, he will fill their boots with sweet treasures. If they have been naughty, they will receive virgács, small twigs that have been spray painted gold and bound together with red decorative ribbon. Actually most children make both the naughty list and the nice list so their boots are filled with a mixture of sweet and the not so sweet including virgács, a subtle reminder to be good.

I especially like this illustration from Wikipedia that features Mikulás (Saint Nickolas) and Krampusz:

I like that it is 150 years old. It is from 1865, several decades before any of my husband’s grandparents left Hungary to live in the United States. Under the chair is a little boy hiding from Krampusz. I like to imagine he’s related to my husband some way. The little girl in the illustration must have been much better behaved that year because she is not afraid enough to need to hide.

Since everyone has been at least a little bit naughty this year, here is virgács for you to put in your shoes tonight, too.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here is a little more about the number 701:

26² + 5² = 701 so it is the hypotenuse of the primitive Pythagorean triple 260-651-701 which can be calculated using 2(26)(5), 26² – 5², 26² + 5².

701 is the sum of three consecutive prime numbers: 229 + 233+ 239.

701 is a palindrome in several bases:

• 10301 BASE 5; note that 1(625) + 0(125) + 3(25) + 0(5) + 1(1) = 701.
• 858 BASE 9; note that 8(81) + 5(9) + 8(1) = 701.
• 1F1 BASE 20; note that F is equivalent to 15 in base 10, and 1(400) + 15(20) + 1(1) = 701.
• 131 BASE 25; note that 1(625) + 3(25) + 1(1) = 701

Stetson.edu informs us that 1^0 + 2^1 + 3^2 + 4^3 + 5^4 = 701.

Here’s another way we know that 701 is a prime number: Since 701 ÷ 4 has a remainder of 1, and 701 can be written as the sum of two squares that have no common prime factors (26² + 5² = 701), then 701 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √701 ≈ 26.4. Since 701 is not divisible by 5, 13, or 17, we know that 701 is a prime number.

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691 and Level 3

• 691 is a prime number.
• Prime factorization: 691 is prime and cannot be factored.
• The exponent of prime number 691 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 691 has exactly 2 factors.
• Factors of 691: 1, 691
• Factor pairs: 691 = 1 x 691
• 691 has no square factors that allow its square root to be simplified. √691 ≈ 26.2868789.

How do we know that 691 is a prime number? If 691 were not a prime number, then it would be divisible by at least one prime number less than or equal to √691 ≈ 26.3. Since 691 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 691 is a prime number.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here’s a little more about 691:

691 is the sum of the thirteen prime numbers from 29 to 79.

691 is equivalent to palindrome 171 in BASE 23. Note that 1(23²) + 7(23) + 1(1) = 691.

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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 10.)  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.

683 is the 4th Wagstaff Prime

683 is the sum of the five prime numbers from 127 to 149. Can you name those five prime numbers?

Like the number before it, 683 has a relationship with the number 11:

(2^11 + 1)/3 = 683. This relationship makes 683 the 4th Wagstaff Prime number. (Notice that 11 is the 4th odd prime number.)

2 raised to an odd prime number has produced many Wagstaff Prime numbers, but not always. For example (2^29 + 1)/3 is not a prime number.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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• 683 is a prime number.
• Prime factorization: 683 is prime and cannot be factored.
• The exponent of prime number 683 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 683 has exactly 2 factors.
• Factors of 683: 1, 683
• Factor pairs: 683 = 1 x 683
• 683 has no square factors that allow its square root to be simplified. √683 ≈ 26.13427.

How do we know that 683 is a prime number? If 683 were not a prime number, then it would be divisible by at least one prime number less than or equal to √683 ≈ 26.1. Since 683 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 683 is a prime number.

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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.