Multiplication table

97 and Will This Six-legged Puzzle Bug You?

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  • 97 is a prime number.
  • Prime factorization: 97 is prime.
  • The exponent of prime number 97 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 97 has exactly 2 factors.
  • Factors of 97: 1, 97
  • Factor pairs: 97 = 1 x 97
  • 97 has no square factors that allow its square root to be simplified. √97 ≈ 9.8488578

How do we know that 97 is a prime number? If 97 were not a prime number, then it would be divisible by at least one prime number less than or equal to √97 ≈ 9.8. Since 97 cannot be divided evenly by 2, 3, 5, or 7, we know that 97 is a prime number.

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

2014-16 Level 2

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

2014-16 Level 2 Factors

96 and Level 1

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Today’s Puzzle:

Write the numbers from 1 to 12 in both the top row and the first column so that this puzzle will function like a multiplication table:

2014-16 Level 1

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

Factors of 96:

96 is a composite number. 96 = 1 x 96, 2 x 48, 3 x 32, 4 x 24, 6 x 16, or 8 x 12. Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96. Prime factorization: 96 = 2 x 2 x 2 x 2 x 2 x 3, which can also be written as 2⁵ x 3.

Sometime 96 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has many other factors, we only use 8 x 12 in the puzzles.

Sum-Difference Puzzle:

6 has two factor pairs. One of those factor pairs adds up to 5 and the other one subtracts to 5. Can you place its factors in the correct boxes in the first puzzle?

96 has six factor pairs. One of those factor pairs adds up to 20, and the another subtracts to 20. If you can identify those two factor pairs, then you can solve the second puzzle!

The second puzzle really is just the first puzzle in disguise. Why would I say that?

Solution to This Week’s Puzzle:

2014-16 Level 1 Factors

95 and Level 6

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95 is a composite number. 95 = 1 x 95 or 5 x 19. The factors of 95: 1, 5, 19, 95. Prime factorization: 95 = 5 x 19.

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

2014-15 Level 6

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-14

2014-15 Level 6 Logic

94 and Level 5

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94 is a composite number. 94 = 1 x 94 or 2 x 47. The factors of 94: 1, 2, 47, 94. Prime factorization: 94 = 2 x 47.

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

2014-15 Level 5

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-14

2014-15 Level 5 Logic

93 and Level 4

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93 is a composite number. 93 = 1 x 93 or 3 x 31. The factors of 93: 1, 3, 31, 93. Prime factorization: 93 = 3 x 31.

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

2014-15 Level 4

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-14

2014-15 Level 4 Logic

92 and Level 3

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92 is a composite number. 92 = 1 x 92, 2 x 46, or 4 x 23.  Factors of 92: 1, 2, 4, 23, 46, 92. Prime factorization: 92 = 2 x 2 x 23 which can also be written 92 = 2² x 23.

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

2014-15 Level 3

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-14

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:

2014-15 Level 3 Factors

90 and Level 2

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A right angle is 90°.

Pythagorean triples are three whole number sides that form a right triangle. In one such triangle, 54-72-90, the longest side, 90, is opposite the 90° angle!

In seven other right triangles, side 90 is adjacent to the 90° angle:

  • 48-90-102
  • 56-90-106
  • 90-120-150
  • 90-216-234
  • 90-400-410
  • 90-672-678
  • 90-2024-2026

90 is a composite number. 90 = 1 x 90, 2 x 45, 3 x 30, 5 x 18, 6 x 15, and 9 x 10. Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. Prime factorization: 90 = 2 × 3 × 3 × 5 which can also be written 2 × 3² × 5.

Sometimes 90 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we only use 90 = 9 x 10 in the puzzles.

2014-15 Level 2

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-14

2014-15 Level 2 Factors

89 and Level 1

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  • 89 is a prime number.
  • Prime factorization: 89 is prime.
  • The exponent of prime number 89 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 89 has exactly 2 factors.
  • Factors of 89: 1, 89
  • Factor pairs: 89 = 1 x 89
  • 89 has no square factors that allow its square root to be simplified. √89 ≈ 9.43398

How do we know that 89 is a prime number? If 89 were not a prime number, then it would be divisible by at least one prime number less than or equal to √89 ≈ 9.4. Since 89 cannot be divided evenly by 2, 3, 5, or 7, we know that 89 is a prime number.

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

2014-15 Level 1

 

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-14

2014-15 Level 1 Factors

88 and April Showers

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88 is a composite number. 88 = 1 x 88, 2 x 44, 4 x 22, and 8 x 11. Factors of 88: 1, 2, 4, 8, 11, 22, 44, 88. Prime factorization: 88 = 2 x 2 x 2 x 11 which can also be written 88 = 2³ x 11.

Sometimes 88 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we only use 88 = 8 x 11 in the puzzles.

2014-14 Level 6

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-07

Getting started logically on this puzzle is a little complicated, and I may have abbreviated too much in the chart below, so here is an explanation in regular English: The clue 14 has to use one of the 2s, so the clue 4 cannot be 2 x 2. Since 4 will be 4 x 1, and thus using one of the 4s, 16 cannot be 4 x 4, but must be 2 x 8. Since 14 and 16 use both of the 2s, 10 cannot be 2 x 5 and must be 1 x 10. Finally that means that the only clue on the puzzle that has 5 as the factor is 35, so the blank space must belong to 5 as well.

2014-14 Level 6 Logic

87 and Level 5

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8 + 7 = 15, a multiple of 3, so 87 is divisible by 3.

87 is a composite number. 87 = 1 x 87 or 3 x 29. Factors of 87: 1, 3, 29, 87. Prime factorization: 87 = 3 x 29.

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

2014-14 Level 5

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-07

2014-14 Level 5 Logic