A Multiplication Based Logic Puzzle

Archive for the ‘Level 4 Puzzle’ Category

955 and Level 4

This level 4 puzzle has ten clues and two of them are the same number! No worries! There is still only one way to write the numbers from 1 to 10 in the first column and the top row so that this puzzle will function like a multiplication table. Can you figure out where to place those numbers?

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

Here’s some information about the number 955:

It is the hypotenuse of a Pythagorean triple:
573-764-955 which is (3-4-5) times 191

It is a palindrome in two other bases:
32323 in BASE 4 because 3(4⁴) + 2(4³) + 3(4²) + 2(4¹) + 3(4⁰) = 955
353 BASE 17 because 3(17²) + 5(17¹) + 3(17⁰) = 955

  • 955 is a composite number.
  • Prime factorization: 955 = 5 × 191
  • 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 955 has exactly 4 factors.
  • Factors of 955: 1, 5, 191, 955
  • Factor pairs: 955 = 1 × 955 or 5 × 191
  • 955 has no square factors that allow its square root to be simplified. √955 ≈ 30.90307

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947 and Level 4

If you know how to multiply and divide, then you can solve this puzzle. Just use logic to find the factors from 1 to 12 that go in the first column and the top row. Go ahead give it a try!

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

Now here’s a little about the number 947:

947 is a prime number that can be written as the sum of seven consecutive prime numbers:
113 + 127 + 131 + 137 + 139 + 149 + 151 = 947

947 is a palindrome in three other bases:
3B3 BASE 16 (B is 11 base 10), because 3(16²) + 11(16¹) + 3(16⁰) = 947
232 BASE 21 because 2(21²) + 3(21¹) + 2(21⁰) = 947
1L1 BASE 22 (L is 21 BASE 10), because 1(22²) + 21(22¹) + 1(22⁰) = 947

  • 947 is a prime number.
  • Prime factorization: 947 is prime.
  • The exponent of prime number 947 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 947 has exactly 2 factors.
  • Factors of 947: 1, 947
  • Factor pairs: 947 = 1 × 947
  • 947 has no square factors that allow its square root to be simplified. √947 ≈ 30.773365

How do we know that 947 is a prime number? If 947 were not a prime number, then it would be divisible by at least one prime number less than or equal to √947 ≈ 30.8. Since 947 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 947 is a prime number.

936 and Level 4

Will this puzzle be smooth sailing for you? You won’t know until you give it a try!

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

936 is the 12th pentagonal pyramidal number because 12²(12 + 1)/2 = 936.

936 is the sum of the twelve prime numbers from 53 to 103.

30² + 6² = 936, so 936 is the hypotenuse of a Pythagorean triple:
360-864-936 which is (5-12-13) times 72, and
can be calculated from 2(30)(6), 30² – 6², 30² + 6²

936 is palindrome 12221 in BASE 5 because 1(5⁴) + 2(5³) + 2(5²) + 2(5¹) + 1(5º) = 936
and palindrome QQ in BASE 35 (Q is 26 base 10), because 26(35) + 26(1) = 26(36) = 936
It is also Q0 in BASE 36 because 26(36) + 0(1) = 936

  • 936 is a composite number.
  • Prime factorization: 936 = 2 × 2 × 2 × 3 × 3 × 13, which can be written 936 = 2³ × 3² × 13
  • The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 936 has exactly 24 factors.
  • Factors of 936: 1, 2, 3, 4, 6, 8, 9, 12, 13, 18, 24, 26, 36, 39, 52, 72, 78, 104, 117, 156, 234, 312, 468, 936
  • Factor pairs: 936 = 1 × 936, 2 × 468, 3 × 312, 4 × 234, 6 × 156, 8 × 117, 9 × 104, 12 × 78, 13 × 72, 18 × 52, 24 × 39¸ or 26 × 36
  • Taking the factor pair with the largest square number factor, we get √936 = (√36)(√26) = 6√26 ≈ 30.594117.

928 Halloween Cat

This cat has arrived just in time for Halloween. Find the factors that go with the clues in the grid to make this Halloween Cat puzzle a multiplication table:

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

Now let me tell you a little about the number 928.

It is the sum of four consecutive prime numbers:
227 + 229 + 233 + 239 = 928
and the sum of eight other consecutive prime numbers:
101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 928

It is the sum of two squares:
28² + 12² = 928
That means 928 is the hypotenuse of a Pythagorean triple:
640-672-928 which is 28² – 12², 2(28)(12), 28² + 12²

Here’s how 928 looks in a few other bases:
It’s 565 in BASE 13, because 5(169) + 6(13) + 5(1) = 928.
It’s 4A4 in BASE 14 (A is 10 base 10), because 4(196) + 10(14) + 4(1) = 928.
It’s TT in BASE 31 (T is 29 base 10), because 29(31) + 29(1) = 29(32) = 928.
It’s T0 in BASE 32, because 29(32) = 928.

  • 928 is a composite number.
  • Prime factorization: 928 = 2 × 2 × 2 × 2 × 2 × 29, which can be written 732 = 2⁵ × 29
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 928 has exactly 12 factors.
  • Factors of 928: 1, 2, 4, 8, 16, 29, 32, 58, 116, 232, 464, 928
  • Factor pairs: 928 = 1 × 928, 2 × 464, 4 × 232, 8 × 116, 16 × 58, or 29 × 32
  • Taking the factor pair with the largest square number factor, we get √928 = (√16)(√58) = 4√58 ≈ 30.463092423.

920 Witches’ Cauldron

“Double, double toil and trouble;
Fire burn, and caldron bubble.”

What besides “eye of newt” goes in witches’ cauldrons? The list includes some horrifying ingredients that you can read here from one scene from Shakeaspeare’s play, MacBeth.

Instead of putting “Eye of newt, and toe of frog, Wool of bat, and tongue of dog” and so forth in today’s Halloween cauldron puzzle, I just put a bunch of asterisks.

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

“Double, double toil and trouble;
Fire burn, and caldron bubble.”

Double 115 is 230.

Double 230 is 460.

Double 460 is 920, today’s post number.

920 is the hypotenuse of a Pythagorean triple:
552-736-920 which is (3-4-5) times 184.

920 is palindrome 767 in BASE 11 because 7(121) + 6(11) + 7(1) = 920

  • 920 is a composite number.
  • Prime factorization: 920 = 2 × 2 × 2 × 5 × 23, which can be written 920 = 2³ × 5 × 23
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 920 has exactly 16 factors.
  • Factors of 920: 1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920
  • Factor pairs: 920 = 1 × 920, 2 × 460, 4 × 230, 5 × 184, 8 × 115, 10 × 92, 20 × 46, or 23 × 40
  • Taking the factor pair with the largest square number factor, we get √920 = (√4)(√230) = 2√230 ≈ 20.331501776.

 

909 and Level 4

909 is a palindrome in base 10 because 9(100) + 0(10) + 9(1) = 909.

It is also palindrome 757 in BASE 11 because 7(121) + 5(11) + 7(1) = 909.

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

Here’s a little more about the number 909:

30² + 3² = 909

909 is the hypotenuse of Pythagorean triple 180-891-909, which is 9 times (20-99-101) and can be calculated from 2(30)(3), 30² – 3², 30² + 3²

As you know, 909 is made with 9’s and 0’s. Stetson.edu informs us that 909 times 2, 3, 4, 5, 6, 7, 8, or 9 do NOT contain even one 9 or 0. That’s a little spooky, but see for yourself:

  • 909 is a composite number.
  • Prime factorization: 909 = 3 × 3 × 101, which can be written 909 = 3² × 101
  • 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 909 has exactly 6 factors.
  • Factors of 909: 1, 3, 9, 101, 303, 909
  • Factor pairs: 909 = 1 × 909, 3 × 303, or 9 × 101
  • Taking the factor pair with the largest square number factor, we get √909 = (√9)(√101) = 3√101 ≈ 30.14962686

901 and Level 4

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

901 is the 25th centered triangular number because (23×24 + 24×25 + 25×26)/2 = 901. That is the same as saying that 901 is the sum of the 23rd, the 24th, and the 25th triangular numbers.

901 is the sum of two squares two different ways:

  • 30² + 1² = 901
  • 26² + 15² = 901

901 is the hypotenuse of FOUR Pythagorean triples:

  • 60-899-901, calculated from 2(30)(1), 30² – 1², 30² + 1²
  • 424-795-901, which is (8-15-17) times 53
  • 451-780-901, calculated from 26² – 15², 2(26)(15), 26² + 15²
  • 476-765-901, which is 17 times (28-45-53)

Two of those were primitives. That can only happen because ALL of 901’s prime factors are Pythagorean triple hypotenuses.

  • 901 is a composite number.
  • Prime factorization: 901 = 17 × 53
  • 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 901 has exactly 4 factors.
  • Factors of 901: 1, 17, 53, 901
  • Factor pairs: 901 = 1 × 901 or 17 × 53
  • 901 has no square factors that allow its square root to be simplified. √901 ≈ 30.016662

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