A Multiplication Based Logic Puzzle

Posts tagged ‘prime’

953 and Level 2

This level 2 puzzle is only a tiny bit more difficult than a level 1 puzzle is. Start by finding the common factors of 4, 12, 40, and 28. The common factors are 1, 2, and 4, but 4 is the only one that works for the puzzle because we aren’t allowed to put factors like 14 or 28 in the top row. We are only allowed to write factors from 1 to 10 in the first column or the top row. Give this puzzle a try. I’m confident you can solve it!

Print the puzzles or type the solution in this excel file: 10-factors-951-958

Prime number 953 is the sum of the 21 prime numbers from 7 to 89.

28² + 13² = 953, so 953 is the hypotenuse of a Pythagorean triple:
615-728-953 calculated from 28² – 13², 2(28)(13), 28² + 13²

953 is a palindrome in base 11 and base 28:
797 in BASE 11 because 7(11²) + 9(11¹) + 7(11⁰) = 953
161 in BASE 28 because 1(28²) + 6(28¹) + 1(28⁰) = 953

953 × 19 × 3 = 54321, making 953 its biggest prime factor. Thank you Stetson.edu for that fun fact.

  • 952 is a prime number.
  • Prime factorization: 952 is prime.
  • The exponent of prime number 952 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 952 has exactly 2 factors.
  • Factors of 952: 1, 952
  • Factor pairs: 952 = 1 × 952
  • 952 has no square factors that allow its square root to be simplified. √952 ≈ 30.854497

How do we know that 952 is a prime number? If 952 were not a prime number, then it would be divisible by at least one prime number less than or equal to √952 ≈ 30.9. Since 952 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 952 is a prime number.

Here’s another way we know that 952 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 13² = 952 with 28 and 13 having no common prime factors, 952 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √952 ≈ 30.9. Since 952 is not divisible by 5, 13, 17, or 29, we know that 952 is a prime number.

 

 

 

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941 and Level 6

Today daylight savings time ends in the United States. That gives you an extra hour to sleep or do whatever you want. Perhaps you could spend part of your extra hour solving today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-932-941

941 is a prime number that is also the sum of consecutive prime numbers two different ways:
179 + 181 + 191 + 193 + 197 = 941; that’s five consecutive primes
311 + 313 + 317 = 941; that’s three consecutive primes

29² + 10² = 941 That means 941 is the hypotenuse of a Pythagorean triple:
580-741-941 which can be calculated from 2(29)(10), 29² – 10², 29² + 10²

941 is also palindrome 575 in BASE 13 because 5(13²) + 7(13¹) + 5(13⁰) = 941

  • 941 is a prime number.
  • Prime factorization: 941 is prime.
  • The exponent of prime number 941 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 941 has exactly 2 factors.
  • Factors of 941: 1, 941
  • Factor pairs: 941 = 1 × 941
  • 941 has no square factors that allow its square root to be simplified. √941 ≈ 30.6757233

How do we know that 941 is a prime number? If 941 were not a prime number, then it would be divisible by at least one prime number less than or equal to √941 ≈ 30.7. Since 941 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 941 is a prime number.

Here’s another way we know that 941 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 10² = 941 with 29 and 10 having no common prime factors, 941 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √941 ≈ 30.7. Since 941 is not divisible by 5, 13, 17, or 29, we know that 941 is a prime number.

 

929 Little Green Monster

Here’s a little green monster just in time for Halloween. It’s a level 5 so it might be a little scary. Just don’t write any of the factors in the first column or top row unless you know for sure that factor belongs where you are putting it. Use logic and not guessing, and you’ll handle this little green monster just fine.

Print the puzzles or type the solution on this excel file: 12 factors 923-931

929 is the sum of nine consecutive prime numbers:
83  + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 929

23² + 20² = 929, so 929 is the hypotenuse of a Pythagorean triple:
129-920-929 which is 23² – 20², 2(23)(20), 23² + 20²

Obviously 929 is a palindrome in base 10.

It is also palindrome 131 in BASE 29 because 1(29²) + 3(29) + 1(1) = 929.

  • 929 is a prime number.
  • Prime factorization: 929 is prime.
  • The exponent of prime number 929 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 929 has exactly 2 factors.
  • Factors of 929: 1, 929
  • Factor pairs: 929 = 1 × 929
  • 929 has no square factors that allow its square root to be simplified. √929 ≈ 30.4795013

How do we know that 929 is a prime number? If 929 were not a prime number, then it would be divisible by at least one prime number less than or equal to √929 ≈ 30.4. Since 929 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 929 is a prime number.

Why Prime Number 919 is the 18th Centered Hexagonal Number

919 is a prime number, but if you had 919 little squares, they could be formed into this fabulous shape:

This hexagon is made from 18 concentric hexagons using the pattern yellow, green, blue, purple, red, and orange repeated. You can easily count that there are 3 × 6 hexagons. (Yes, that’s counting the yellow square in the center as a hexagon because 1 is the first centered hexagonal number.) Here’s why prime number 919 is a centered hexagonal number:

919 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60 + 66 + 72 + 78 + 84 + 90 + 96 + 102, the number of squares contained in those concentric hexagons listed in order from smallest to largest . Thus,
919 = 1 + 6(1 + 2 + 3 + 4 +  5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17) = 1 + 6(153)

919 is a centered hexagonal number because 919 – 1 is 918. What was special about 918? Well, consecutive numbers, 17 and 18, are two of its factors. That made the 17th triangular number a factor of 918. Because 918 is 6 times a triangular number (153), the next number, 919, is a centered hexagonal number.

919 is the 18th centered hexagonal number because 817 (the 17th centered hexagonal number) plus 6(17) = 919.

919 is the 18th centered hexagonal number because 630 (the 18th hexagonal number) plus 17² = 919:

919 is also the 18th centered hexagonal number because 18³ – 17³ = 919. Even though the difference of two cubes can always be factored, 919 is still a prime number because
18³ – 17³ = (18 – 17)(18² + (17)(18) + 17²) = (1)(919)

919 is not only the 18th centered hexagonal number, but it is a palindrome in base 10 and two other bases:

414 in BASE 15

171 in BASE 27

919 uses its same digits, 199, in BASE 26

  • 919 is a prime number.
  • Prime factorization: 919 is prime.
  • The exponent of prime number 919 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 919 has exactly 2 factors.
  • Factors of 919: 1, 919
  • Factor pairs: 919 = 1 × 919
  • 919 has no square factors that allow its square root to be simplified. √919 ≈ 30.31501

How do we know that 919 is a prime number? If 919 were not a prime number, then it would be divisible by at least one prime number less than or equal to √919 ≈ 30.3. Since 919 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 919 is a prime number.

 

911 Mystery Level Puzzle

911 is a prime number that is also the sum of three consecutive primes:

  • 293 + 307 + 311 = 911

911 is 191 in BASE 26 because 1(26²) + 9(26) + 1(1) = 911

911 is palindrome 12121 in BASE 5 because 1(5⁴) + 2(5³) + 1(5²) + 2(5) + 1(1) = 911

Print the puzzles or type the solution on this excel file: 12 factors 905-913

  • 911 is a prime number.
  • Prime factorization: 911 is prime.
  • The exponent of prime number 911 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 911 has exactly 2 factors.
  • Factors of 911: 1, 911
  • Factor pairs: 911 = 1 × 911
  • 911 has no square factors that allow its square root to be simplified. √911 ≈ 30.1827765

How do we know that 911 is a prime number? If 911 were not a prime number, then it would be divisible by at least one prime number less than or equal to √911 ≈ 30.2. Since 911 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 911 is a prime number.

 

 

 

 

907 and Level 3

88 is 8 × 11, and 24 is 2 × 12, 3 × 8, or 4 × 6. Those are all their factor pairs in which both factors are less than or equal to 12. There is only one number that is a common factor for both 88 and 24. Write that number in the top cell of the first column and the rest of 88’s factor pair directly above 88 and the rest of 24’s factor pair directly above 24 in the top row.

Next think of a factor pair for 80 in which both factors are less than or equal to 12. You probably thought of 8 × 10, the only factor pair that qualifies. The first column already has an 8, so this 8 must go in the top row above 80. Write 10 in the first column.

The next row doesn’t have a clue, but you already have enough information to write what number must go in the first column. (Hint: it is a number that is already in the top row and can’t go in any other cell in the first column.) If you cannot figure out what goes in this cell, skip that row until later, and figure out what goes in the next cells continuing from the top cell of the first column to the bottom cell. You will fill out the top row at the same time, but each factor 1- 12 will be written above its appropriate clue instead of in order from left to right. Good luck!

Print the puzzles or type the solution on this excel file: 12 factors 905-913

907 is the first prime number since 887. We will not have to wait nearly as long for the next prime number. It will be 911.

907 is palindrome 32023 in BASE 4 because 3(4⁴) + 2(4³) + 0(4²) + 2(4¹) + 3(4º) = 907.

  • 907 is a prime number.
  • Prime factorization: 907 is prime.
  • The exponent of prime number 907 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 907 has exactly 2 factors.
  • Factors of 907: 1, 907
  • Factor pairs: 907 = 1 × 907
  • 907 has no square factors that allow its square root to be simplified. √907 ≈ 30.11644

How do we know that 907 is a prime number? If 907 were not a prime number, then it would be divisible by at least one prime number less than or equal to √907 ≈ 30.1. Since 907 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 907 is a prime number.

887 is the Last Prime Number for a While!

The last prime number before 887 was 883.

The next prime number won’t be until 907. Wow!

907 – 887 = 20. That’s the largest gap between prime numbers so far!

887 is also a palindrome in a few other bases:

  • 31313 BASE 4, because 3(4⁴) + 1(4³) + 3(4²) + 1(4) + 3(1) = 887
  • 737 BASE 11, because 7(11²) + 3(11) + 7(1) = 887
  • 313 BASE 17, because 3(17²) + 1(17) + 3(1) = 887

Those patterns of 3’s and 1’s for two different bases surprised me!

  • 887 is a prime number.
  • Prime factorization: 887 is prime.
  • The exponent of prime number 887 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 887 has exactly 2 factors.
  • Factors of 887: 1, 887
  • Factor pairs: 887 = 1 × 887
  • 887 has no square factors that allow its square root to be simplified. √887 ≈ 29.782545

How do we know that 887 is a prime number? If 887 were not a prime number, then it would be divisible by at least one prime number less than or equal to √887 ≈ 29.8. Since 887 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 887 is a prime number.

 

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