divisors

80 and A Puzzled Cat

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WordPress measures how many posts are written about certain topics each month and then they publish their findings. The results for March were published in the march-hot-list/. Cat lovers were informed, “your beloved felines yet again got trounced by their canine frenemies. Dogs: 10,589; cats: 6,401. Is that a big, defiant “meow” we hear? Maybe you’ll do better in April.”

I have two sisters who truly love their cats. Although I personally prefer neither pet over the other, I hate to see cats so clearly in the “underdog” role. Today’s puzzle looks a teeny tiny bit like the head of a cat, so I’m tagging it with the word “cats.” Of course, this post alone will not be enough to make much of a difference in April’s statistics.

2014-13 Level 4

This week’s puzzles and last week’s solutions: 10 Factors 2014-03-31

80 is a composite number. 80 = 1 x 80, 2 x 40, 4 x 20, 5 x 16, and 8 x 10. Factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80. Prime factorization: 80 = 2 x 2 x 2 x 2 x 5 which can also be written 80 = 2⁴ x 5.

Sometimes 80 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact we use is 8 x 10 = 80.

2014-13 Level 4 Logic

79 and Level 3

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2014-13 Level 3

This week’s puzzles and last week’s solutions: 10 Factors 2014-03-31

  • 79 is a prime number.
  • Prime factorization: 79 is prime.
  • The exponent of prime number 79 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 79 has exactly 2 factors.
  • Factors of 79: 1, 79
  • Factor pairs: 79 = 1 x 79
  • 79 has no square factors that allow its square root to be simplified. √79 ≈ 8.8881944

How do we know that 79 is a prime number? If 79 were not a prime number, then it would be divisible by at least one prime number less than or equal to √79 ≈ 8.8. Since 79 cannot be divided evenly by 2, 3, 5, or 7, we know that 79 is a prime number.

79 is never a clue in the FIND THE FACTORS puzzles.

A Logical Approach to find the factors: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column and factor row.  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the table one row at a time as you go:

2014-13 Level 3 Factors

78 and Patterns of Ideas (Hardy)

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78 is a composite number. 78 = 1 x 78, 2 x 39, 3 x 26, or 6 x 13. Factors of  78: 1, 2, 3, 6, 13, 26, 39, 78. Prime factorization: 78 = 2 x 3 x 13.

78 is never a clue in the FIND THE FACTORS 1-10 or 1-12 puzzles.

Here’s a sample of a very creative cartoon by Marek Bennett. I especially like this one because it features a quote from G. H. Hardy, a hero of mine. Because of Hardy, the world knows about Ramanujan, possibly the greatest mathematician of the 20th century.

77 and Level 2

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2014-13 Level 2

This week’s puzzles and last week’s solutions: 10 Factors 2014-03-31

77 is a composite number. 77 = 1 x 77 or 7 x 11. Factors of 77: 1, 7, 11, 77. Prime factorization: 77 = 7 x 11.

Sometimes 77 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, the puzzle only uses 7 x 11 = 77.

2014-13 Level 2 Factors

76 and Level 1

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2014-13 Level 1

This week’s puzzles and last week’s solutions: 10 Factors 2014-03-31

76 is a composite number. 76 = 1 x 76, 2 x 38, or 4 x 19. Factors of 76: 1, 2, 4, 19, 38, 76, Prime factorization: 76 = 2 x 2 x 19, which can also be written 2² x 19.

76 is never a clue in the FIND THE FACTORS puzzles.

2014-13 Level 1 Factors

75 and Level 6

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2014-12 Level 6

This week’s puzzles for you to solve

75 is a composite number. 75 = 1 x 75, 3 x 25, or 5 x 15. Factors of 75: 1, 3, 5, 15, 25, 75. Prime factorization: 75 = 3 x 5 x 5, which can also be written 75 = 3 x 5².

75 is never a clue in the FIND THE FACTORS puzzles.

75 is in this wonderful pattern. Can you tell what the pattern is?

2014-12 Level 6 Logic

74 and Level 5

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2014-12 Level 5

This week’s puzzles for you to solve

74 is a composite number. 74 = 1 x 74 or 2 x 37. Factors of 74: 1, 2, 37, 74. Prime factorization: 74 = 2 x 37.

74 is never a clue in the FIND THE FACTORS puzzles.

2014-12 Level 5 Logic

73 and Once

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In Hungarian, “Multiplication Table” and “Times Table” are the same expression as “Once Upon a Time”. I very much enjoyed learning that when I went to the school of fairies today where even butterflies can learn to multiply. The author translates a sweet poem from Hungarian into English. I hope you will read the poem and remember its encouraging words whenever you try to solve one of my multiplication table puzzles or any other task that challenges you in life.

  • 73 is a prime number.
  • Prime factorization: 73 is prime.
  • The exponent of prime number 73 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 73 has exactly 2 factors.
  • Factors of 73: 1, 73
  • Factor pairs: 73 = 1 x 73
  • 73 has no square factors that allow its square root to be simplified. √73 ≈ 8.5440037

How do we know that 73 is a prime number? If 73 were not a prime number, then it would be divisible by at least one prime number less than or equal to √73 ≈ 8.5. Since 73 cannot be divided evenly by 2, 3, 5, or 7, we know that 73 is a prime number.

73 is never a clue in the FIND THE FACTORS puzzles.

73 is included in this list of prime numbers:

 

72 and Level 4

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2014-12 Level 4

This week’s puzzles for you to solve

Here are some facts about the number 72:

72 is a composite number. 72 = 1 x 72, 2 x 36, 3 x 24, 4 x 18, 6 x 12, or 8 x 9. Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Prime factorization: 72 = 2 x 2 x 2 x 3 x 3, which can also be written 2³ x 3²

Sometimes 72 is a clue in the Find the Factors puzzles. Even though it has many other factors, we use only 8 x 9 for the FIND THE FACTORS 1-10 puzzles and 8 x 9 or 6 x 12 for the FIND THE FACTORS 1-12 puzzles.

2014-12 Level 4 Logic

71 and Level 3

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2014-12 Level 3

This week’s puzzles for you to solve

Here’s a little about the number 71:

  • 71 is a prime number.
  • Prime factorization: 71 is prime.
  • The exponent of prime number 71 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 71 has exactly 2 factors.
  • Factors of 71: 1, 71
  • Factor pairs: 71 = 1 x 71
  • 71 has no square factors that allow its square root to be simplified. √71 ≈ 8.4261

How do we know that 71 is a prime number? If 71 were not a prime number, then it would be divisible by at least one prime number less than or equal to √71 ≈ 8.4. Since 71 cannot be divided evenly by 2, 3, 5, or 7, we know that 71 is a prime number.

71 is never a clue in the FIND THE FACTORS puzzles.

A Logical Approach to the Solution of the puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column and factor row.  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the table as you go:

2014-12 Level 3 Factors