A Multiplication Based Logic Puzzle

Archive for the ‘Level 6 Puzzle’ Category

1018 and Level 6

Level 6 puzzles can be tricky to solve, but I promise that you can still solve this one using logic and knowledge of the basic multiplication table. Just write the numbers from 1 to 12 in both the first column and the top row so that the puzzle is like a partially filled out multiplication table with the factors in a different order. Like always, there is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Look at these interesting facts about the number 1018:

27² + 17² = 1018
That means that 1018 is the hypotenuse of a Pythagorean triple:
440-918-1018 calculated from 27² – 17², 2(27)(17), 27² + 17²

It also means that (44² – 10²)/2 = 1018
Note that 27 + 17 = 44 and 27 – 17 = 10

1018 is full house 33322 in BASE 4 because 3(4⁴) + 3(4³) + 3(4²) + 2(4¹) + 2(4⁰) = 3(256 + 64 + 16) + 2(4 + 1) = 1018

Since 1018 is the sum of odd squares, it is divisible by 2. Since those odd squares have no common prime factors, you only have to check to see if 1018 is divisible by any Pythagorean triple hypotenuses less than or equal to (√1018)/2 ≈ 15.953. It is not divisible by 5 or 13, therefore 1018 only has two prime factors: 2 and 1018/2.

  • 1018 is a composite number.
  • Prime factorization: 1018 = 2 × 509
  • 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 1018 has exactly 4 factors.
  • Factors of 1018: 1, 2, 509, 1018
  • Factor pairs: 1018 = 1 × 1018 or 2 × 509
  • 1018 has no square factors that allow its square root to be simplified. √1018 ≈ 31.90611

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1007 and Level 6

Some of the clues in this puzzle pair up in the same column or the same row and try to trick you into picking the wrong common factor. Nevertheless, the 10 clues in the puzzle work together to give you the most logical place to start the puzzle. It may be a little difficult to see the logic for this one but stick with it. You’ll figure it out.

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

Here’s a little bit about the number 1007:

1007 is the hypotenuse of a Pythagorean triple:
532-855-1007lwhich is 19 times (28-45-53)

1007 is palindrome 33233 in BASE 4
because 3(4⁴) + 3(4³) + 2(4²) + 3(4¹) + 3(4⁰) = 1007

  • 1007 is a composite number.
  • Prime factorization: 1007 = 19 × 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 1007 has exactly 4 factors.
  • Factors of 1007: 1, 19, 53, 1007
  • Factor pairs: 1007 = 1 × 1007 or 19 × 53
  • 1007 has no square factors that allow its square root to be simplified. √1007 ≈ 31.73326

1001 and Level 6

There is only one multiplication table that has the numbers you see in this puzzle exactly where you see them here. Can you find the factors that create that multiplication table? This is a level 6 puzzle so it won’t be easy, but it can be done by just using logic and the basic 1-12 multiplication facts.

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

1001 is the product of three consecutive prime numbers:
7 × 11 × 13 = 1001

1001 is also the sum of fifteen consecutive prime numbers:
37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 1001

1001 is the 26th pentagonal number.

1001 is the hypotenuse of a Pythagorean triple:
385² + 924² = 1001²

1001 is a palindrome in base 10 and in base 25:
It’s 1F1 in BASE 25 (F is 15 base 10) because 1(625) + 15(25) + 1(1) = 1001

  • 1001 is a composite number.
  • Prime factorization: 1001 = 7 × 11 × 13
  • 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 1001 has exactly 8 factors.
  • Factors of 1001: 1, 7, 11, 13, 77, 91, 143, 1001
  • Factor pairs: 1001 = 1 × 1001, 7 × 143, 11 × 91, or 13 × 77
  • 1001 has no square factors that allow its square root to be simplified. √1001 ≈ 31.63858

992 Christmas Factor Tree

Artificial Christmas trees have to be assembled. Sometimes the assembly is easy, and sometimes it is frustrating.

This Christmas tree puzzle can be solved using LOGIC and an ordinary multiplication table, but there’s a good chance it will frustrate you. Go ahead and try to solve it!

Print the puzzles or type the solution in this excel file: 10-factors-986-992

The number 992 also can make a nice looking, well-balanced factor tree:

992 is the product of two consecutive numbers: 31 × 32 = 992.
Because of that fact, 992 is the sum of the first 31 EVEN numbers:
2 + 4 + 6 + 8 + 10 + . . . + 54 + 56 + 58 + 60 + 62 = 992

992 is palindrome 212 in BASE 22 because 2(22²) + 1(22) + 2(1) = 922. That was a lot of 2’s and 1’s in that fun fact!

  • 992 is a composite number.
  • Prime factorization: 992 = 2 × 2 × 2 × 2 × 2 × 31, which can be written 732 = 2⁵ × 31
  • 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 992 has exactly 12 factors.
  • Factors of 992: 1, 2, 4, 8, 16, 31, 32, 62, 124, 248, 496, 992
  • Factor pairs: 992 = 1 × 992, 2 × 496, 4 × 248, 8 × 124, 16 × 62, or 31 × 32
  • Taking the factor pair with the largest square number factor, we get √992 = (√16)(√62) = 4√62 ≈ 31.49603

983 Candy Cane

Candy canes have been a part of the Christmas season for ages. Here’s a candy cane puzzle for you to try. It’s a level 6 so it won’t be easy, but you will taste its sweetness once you complete it. Go ahead and get started!

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Here’s some information about prime number 983:

983 is the sum of consecutive prime numbers two different ways:
It is the sum of the seventeen prime numbers from 23 to 97.
It is also the sum of the thirteen prime numbers from 47 to 103.

  • 983 is a prime number.
  • Prime factorization: 983 is prime.
  • The exponent of prime number 983 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 983 has exactly 2 factors.
  • Factors of 983: 1, 983
  • Factor pairs: 983 = 1 × 983
  • 983 has no square factors that allow its square root to be simplified. √983 ≈ 31.35283

How do we know that 983 is a prime number? If 983 were not a prime number, then it would be divisible by at least one prime number less than or equal to √983 ≈ 31.4. Since 983 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 983 is a prime number.

977 and Level 6

Can you find the logic that is needed to make these ten clues become a multiplication table with the numbers 1 to 10 in the first column and again in the top row? This is a level 6 puzzle so it should be a little bit of a challenge even for adults. Nevertheless, it probably won’t require too much of your time to solve. Go ahead. Give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-968-977

Prime number 977 is the sum of nine consecutive prime numbers:
89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 = 977

31² + 4² = 977 making 977 the hypotenuse of a Pythagorean triple:
248-945-977 calculated from 2(31)(4), 31² – 4², 31² + 4²

  • 977 is a prime number.
  • Prime factorization: 977 is prime.
  • The exponent of prime number 977 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 977 has exactly 2 factors.
  • Factors of 977: 1, 977
  • Factor pairs: 977 = 1 × 977
  • 977 has no square factors that allow its square root to be simplified. √977 ≈ 31.256999

How do we know that 977 is a prime number? If 977 were not a prime number, then it would be divisible by at least one prime number less than or equal to √977 ≈ 31.3. Since 977 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 977 is a prime number.

Here’s another way we know that 977 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 4² = 977 with 31 and 4 having no common prime factors, 977 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √977 ≈ 31.3. Since 977 is not divisible by 5, 13, 17, or 29, we know that 977 is a prime number.

 

966 Groan! Gotta Loosen My Belt

If you’ve overeaten this Thanksgiving day, you may be in too much pain to start working off all those extra calories. You may just want to loosen your belt and lie down somewhere while you groan about eating so much. Exercising your brain may help you alleviate some of that regret. This puzzle with its Pilgrim belt buckle could be just what you need. It’s a level 6 so it won’t be easy, but you will feel very accomplished if you can solve it.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

Here’s a little information about the number 966:

It is the sum of eight consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 = 966

It is also the sum of two consecutive prime numbers:
467 + 479 = 966

966 is palindrome 686 in BASE 12 because 6(144) + 8(12) + 6(1) = 966

  • 966 is a composite number.
  • Prime factorization: 966 = 2 × 3 × 7 × 23
  • 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 × 2 × 2 × 2 = 16. Therefore 966 has exactly 16 factors.
  • Factors of 966: 1, 2, 3, 6, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966
  • Factor pairs: 966 = 1 × 966, 2 × 483, 3 × 322, 6 × 161, 7 × 138, 14 × 69, 21 × 46, or 23 × 42
  • 966 has no square factors that allow its square root to be simplified. √966 ≈ 31.08054

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