A Multiplication Based Logic Puzzle

Archive for the ‘Level 2 Puzzle’ Category

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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943 and Level 2

Level 2 puzzles aren’t very tricky, but maybe this one is a little bit. Can you write the factors from 1 to 12 in both the first column and the top row so that this puzzle functions as a multiplication table?

Print the puzzles or type the solution in this excel file: 12 factors 942-950

Now I’ll write something about the number 943:

943 is the hypotenuse of a Pythagorean triple:
207-920-943 which is 23 times (9-40-41)

It is also a leg in two primitive Pythagorean triples:
576-943-1105, calculated from 2(32)(9), 32² – 9², 32² + 9²
943-444624-444625, calculated from 472² – 471², 2(472)(471), 472² + 471²

  • 943 is a composite number.
  • Prime factorization: 943 = 23 × 41
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 943 has exactly 4 factors.
  • Factors of 943: 1, 23, 41, 943
  • Factor pairs: 943 = 1 × 943 or 23 × 41
  • 943 has no square factors that allow its square root to be simplified. √943 ≈ 30.708305

934 and Level 2

I hope you are having a happy Halloween at your house. This puzzle is only a level 2. I’m confident that you can find where all the factors from 1 to 10 go in both the first column and the top row.

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

Now let me tell you a little about the number 934:

934 is palindrome 424 in BASE 15 because
4(15²) + 2(15¹) + 4(15º) = 934

  • 934 is a composite number.
  • Prime factorization: 934 = 2 × 467
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 934 has exactly 4 factors.
  • Factors of 934: 1, 2, 467, 934
  • Factor pairs: 934 = 1 × 934 or 2 × 467
  • 934 has no square factors that allow its square root to be simplified. √934 ≈ 30.56141

Here are some mathematical posts I saw on Twitter to celebrate Halloween:

https://platform.twitter.com/widgets.js

Halloween wishes to all mathematical types out there 😱 pic.twitter.com/0Jyk9IBNmp

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

926 Creepy Crawler

You may see lots of crazy creepy-crawling critters this Halloween. Some of them may look very scary.

This level 2 puzzle is really quite tame. Don’t be afraid of it!

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

926 is the sum of six consecutive prime numbers:
139 + 149 + 151 + 157 + 163 + 167 = 926

926 is repdigit 222 in BASE 21 because 2(21²) + 2(21) + 2(1) = 2(463) = 926

926 is also palindrome 1C1 in BASE 25 (C is 12 in base 10)

  • 926 is a composite number.
  • Prime factorization: 926 = 2 × 463
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 926 has exactly 4 factors.
  • Factors of 926: 1, 2, 463, 926
  • Factor pairs: 926 = 1 × 926 or 2 × 463
  • 926 has no square factors that allow its square root to be simplified. √926 ≈ 30.430248

916 Witch’s Hat

Today’s puzzle looks a little like a witch’s hat. Solving it could be bewitching. It contains a lot of perfect squares, and it’s only a level 2. Give it a try! Just make sure you don’t write a number greater than ten in the top row or the first column. Use all the numbers from one to ten in both places. There is only one solution.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

916 looks the same upside-down as it does right-side up, so it is a Strobogrammatic number.

It looks like someone put a spell on that number!

Not only that, in BASE 26, our 916 looks like 196. There is a whole lot of magic going on here!

30² + 4² = 916, so 916 is the hypotenuse of a Pythagorean triple:
240-884-916, which can be calculated from 2(30)(4), 30² – 4², 30² + 4².

  • 916 is a composite number.
  • Prime factorization: 916 = 2 × 2 × 229, which can be written 916 = 2² × 229
  • 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 916 has exactly 6 factors.
  • Factors of 916: 1, 2, 4, 229, 458, 916
  • Factor pairs: 916 = 1 × 916, 2 × 458, or 4 × 229
  • Taking the factor pair with the largest square number factor, we get √916 = (√4)(√229) = 2√229 ≈ 30.2654919

 

 

906 and Level 2

906 looks the same upside-down so it is a strobogrammatic number.

We can tell just by looking at 906 that it is divisible by 1, 2, and 3. That means it can also be evenly divided by 6.

906 is the sum of consecutive prime numbers: 449 + 457 = 906

It is also palindrome 636 in BASE 12 because 6(144) + 3(12) + 6(1) = 906.

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

  • 906 is a composite number.
  • Prime factorization: 906 = 2 × 3 × 151
  • 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 906 has exactly 8 factors.
  • Factors of 906: 1, 2, 3, 6, 151, 302, 453, 906
  • Factor pairs: 906 = 1 × 906, 2 × 453, 3 × 302, or 6 × 151
  • 906 has no square factors that allow its square root to be simplified. √906 ≈ 30.099833887

 

898 and Level 2

27² + 13² = 898. That means that 898 is the hypotenuse of a Pythagorean triple:

  • 560-702-898, which is 2 times (280-351-449).

898 reads the same way frontwards and backwards so it is a palindrome in base 10.

It is also palindrome 747 in BASE 11 because 7(11²) + 4(11) + 7(1) = 898

AND it is palindrome 1G1 in BASE 23 (G is 16 in base 10) because 1(23²) + 16(23) +1(1) =898.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 898 is a composite number.
  • Prime factorization: 898 = 2 × 449
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 898 has exactly 4 factors.
  • Factors of 898: 1, 2, 449, 898
  • Factor pairs: 898 = 1 × 898 or 2 × 449
  • 898 has no square factors that allow its square root to be simplified. √898 ≈ 29.9666

 

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