A Multiplication Based Logic Puzzle

Archive for the ‘Puzzles’ Category

799 A Rose for Your Valentine

Roses are beautiful and make lovely gifts for Valentines or any other occasion. A Native American legend explains why roses have thorns.

The rose on today’s puzzle has thorns because without thorny clue 60 the puzzle would not have a unique solution. You can be sure that 60 will play an important part in using logic to find the solution to this puzzle.

With or without a valentine, love your brain and give the puzzle a try. It won’t be easy, but you should eventually be able to figure it out. My blogging friend, justkinga, has some other suggestions to show YOURSELF some love on Valentine’s Day.

799-puzzle

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

7 + 9 + 9 = 25, a composite number.

7^3 + 9^3 + 9^3 = 1801, a prime number.

Stetson.edu states that 799 is the smallest number whose digits add up to a composite number AND whose digits cubed add up to a prime number.

It may seem like an improbable number fact, but it wasn’t too difficult to verify, and it really is true!

799 is also the smallest number whose digits add up to 25. (The digits of 889 also add up to 25, and its digits cubed also add up to a prime number. Could this be more than a coincidence?)

Here’s more about the number 799:

799 is palindrome 1H1 in BASE 21 (H is 17 base 10). Note that 1(441) + 17(21) + 1(1) = 799.

799 is the hypotenuse of Pythagorean triple 376-705-799 which is 47 times 8-15-17.

  • 799 is a composite number.
  • Prime factorization: 799 = 17 x 47
  • 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 799 has exactly 4 factors.
  • Factors of 799: 1, 17, 47, 799
  • Factor pairs: 799 = 1 x 799 or 17 x 47
  • 799 has no square factors that allow its square root to be simplified. √799 ≈ 266588.

799-factor-pairs

 

 

 

Advertisements

798 Cupid’s Arrow and Target

Here are two puzzles that go together and yet look out of sync. Sometimes cupid’s arrow reaches its target, and sometimes it doesn’t.

798-puzzle

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

Here’s a little about the number 798:

798 is made from three consecutive numbers (7, 8, and 9), so it is divisible by three. The middle number, 8, is not divisible by three, so 798 is NOT divisible by nine.

798 is a palindrome in two bases:

  • 666 BASE 11 because 6(121) + 6(11) + 6(1) = 798
  • 383 BASE 15 because 3(225) + 8(15) + 3(1) = 798

798 is also the sum of two consecutive prime numbers: 397 + 401 = 798.

798 can be written as the sum of three squares four different ways:

  • 26² + 11² + 1² = 798
  • 25² + 13² + 2² =798
  • 23² + 13² + 10² = 798
  • 22² + 17² + 5² = 798

Here is 798’s factoring information:

  • 798 is a composite number.
  • Prime factorization: 798 = 2 x 3 x 7 x 19
  • 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 798 has exactly 16 factors.
  • Factors of 798: 1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798
  • Factor pairs: 798 = 1 x 798, 2 x 399, 3 x 266, 6 x 133, 7 x 114, 14 x 57, 19 x 42, or 21 x 38
  • 798 has no square factors that allow its square root to be simplified. √798 ≈ 28.24889.

798-factor-pairs

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

 

796 and a Valentine

Here’s a puzzle you can print, cut out, and give as a valentine:

796-puzzle

You may know the divisibility rules for these powers of two:

  • If the last digit of a number is divisible by 2, the whole number is divisible by 2.
  • If the last two digits are divisible by 4, the whole number is divisible by 4.
  • If the last three digits are divisible by 8, the whole number is divisible by 8.

But I’m going to apply some other time-saving but possibly more confusing divisibility rules to the number 796:

  • 796 is divisible by 2 because 6 is an even number.
  • 796 is divisible by 4 because even number 6 is NOT divisible by 4, and 9 is an odd number.
  • 796 is NOT divisible by 8 because 96 is divisible by 8, and 7 is an odd number.

Because 796 is divisible by 4 but not by 8, it can be written as the sum of 8 consecutive numbers:

  • 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103 = 796

796 is also the sum of all the prime numbers from 113 to 149:

  • 113 + 127 + 131 + 137 + 139 + 149 = 796

Here is the factoring information for 796:

  • 796 is a composite number.
  • Prime factorization: 796 = 2 x 2 x 199, which can be written 796 = (2^2) x 199
  • 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 796 has exactly 6 factors.
  • Factors of 796: 1, 2, 4, 199, 398, 796
  • Factor pairs: 796 = 1 x 796, 2 x 398, or 4 x 199
  • Taking the factor pair with the largest square number factor, we get √796 = (√4)(√199) = 2√199 ≈ 28.21347.

796-factor-pairs

 

795 a Level 1 Puzzle with ♥

I learned yesterday that pushing ALT on the computer key pad and 3 on the number pad at the same time produces a when the keys are released. I added a few ‘s to today’s puzzle because Valentine’s day is almost here:

795-puzzle

  • 795 is a composite number.
  • Prime factorization: 795 = 3 x 5 x 53
  • 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 795 has exactly 8 factors.
  • Factors of 795: 1, 3, 5, 15, 53, 159, 265, 795
  • Factor pairs: 795 = 1 x 795, 3 x 265, 5 x 159, or 15 x 53
  • 795 has no square factors that allow its square root to be simplified. √795 ≈ 28.195744.

795-factor-pairs

Stetson.edu informs us that if we add up all the factors of 795 we get a number that is a perfect 4th power:

  • 1 + 3 + 5 + 15 + 53 + 159 + 265 + 795 = 1296, which happens to be the 4th power of 6. (Of course 1296 also equals 36 squared.)

795 is the hypotenuse of four Pythagorean triples and thus:

  • 69 – 792 – 795 which is 3 times (23 – 264 – 265)
  • 288 – 741 – 795 which is 3 times (96 – 247 – 265)
  • 420 – 675 – 795 which is 15 times (28 – 45 – 53)
  • 477 – 636 – 795 which is 159 times (3 – 4 – 5)

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

  • 25² + 13² + 1² = 795
  • 25² + 11² + 7² = 795

 

 

794 and Level 6

794 is the hypotenuse of a Pythagorean triple, 456-650-794, so 456² + 650² = 794².

794 is also palindrome 282 in BASE 18. Note that 2(18²) + 8(18) + 2(1) = 794.

Stetson.edu informs us that 1^6 + 2^6 + 3^6 = 1 + 64 + 729 = 794.

794-puzzle

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

————————————–

  • 794 is a composite number.
  • Prime factorization: 794 = 2 x 397
  • 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 794 has exactly 4 factors.
  • Factors of 794: 1, 2, 397, 794
  • Factor pairs: 794 = 1 x 794 or 2 x 397
  • 794 has no square factors that allow its square root to be simplified. √794 ≈ 1780056.

794-factor-pairs

————————————–

793 and Level 5

793 is the sum of two squares TWO different ways!

  • 28² + 3² = 793
  • 27² + 8² = 793

Notice that 793 is 4(198) + 1, and neither 28 and 3 or 27 and 8 have any common prime factors. Could 793 possibly be a prime number?

The answer is no for two reasons:

  1. √793 is about 28.2, so we only need to check to see if 793 is divisible by 5, 13, or 17 (all the 4N+1 prime numbers that are less than 28.) However, 793 ÷ 13 = 61, so  793 isn’t prime.
  2. Also, any number that can be written as the sum of two squares in more than one way is never a prime number.

Both 13 and 61 give a remainder of one when they are divided by four, and 793 is the hypotenuse of FOUR Pythagorean triples:

  • 143-780-793, which is 13 times 11-60-61
  • 168-775-793, a primitive calculated from 2(28)(3), 28² – 3², 28² + 3²
  • 305-732-793, which is 61 times 5-12-13
  • 432-665-793, a primitive calculated from 2(27)(8), 27² – 8², 27² + 8²

793 is also palindrome 191 in BASE 24; note that 1(24²) + 9(24) + 1(1) = 793

Finally Stetson.edu informs us that 793 is 2(397) – 1.

793-puzzle

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

————————————–

  • 793 is a composite number.
  • Prime factorization: 793 = 13 x 61
  • 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 793 has exactly 4 factors.
  • Factors of 793: 1, 13, 61, 793
  • Factor pairs: 793 = 1 x 793 or 13 x 61
  • 793 has no square factors that allow its square root to be simplified. √793 ≈ 28.16025568.

793-factor-pairs

Tag Cloud