divisors

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

86 and Level 4

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86 is a composite number. 86 = 1 x 86 or 2 x 43. Factors of 86: 1, 2, 43, 86. Prime factorization: 86 = 2 x 43.

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

2014-14 Level 4

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

2014-14 Level 4 Logic

85 and Level 3

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2014-14 Level 3

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

85 is a composite number. 85 = 1 x 85 or 5 x 17. Factors of 85: 1, 5, 17, 85. Prime factorization: 85 = 5 x 17.

85 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:

2014-14 Level 3 Factors

84 and Level 2

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Today’s Puzzle:

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

2014-14 Level 2

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

Factors of 84:

84 is a composite number. 84 = 1 x 84, 2 x 42, 3 x 28, 4 x 21, 6 x 14, or 7 x 12. Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Prime factorization: 84 = 2 x 2 x 3 x 7 which can also be written 2² x 3 x 7.

Sometimes 84 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we only use 84 = 7 x 12 in the puzzles.

Sum-Difference Puzzle:

84 has six factor pairs. One of those factor pairs adds up to 25, and another one subtracts to 25. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 84:

84 is prominent in this cool pattern:

Answer to This Week’s Puzzle:

2014-14 Level 2 Factors

83 and Level 1

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2014-14 Level 1

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

  • 83 is a prime number.
  • Prime factorization: 83 is prime.
  • The exponent of prime number 83 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 83 has exactly 2 factors.
  • Factors of 83: 1, 83
  • Factor pairs: 83 = 1 x 83
  • 83 has no square factors that allow its square root to be simplified. √83 ≈ 9.11043

How do we know that 83 is a prime number? If 83 were not a prime number, then it would be divisible by at least one prime number less than or equal to √83 ≈ 9.1. Since 83 cannot be divided evenly by 2, 3, 5, or 7, we know that 83 is a prime number.

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

 2014-14 Level 1 Factors

82 and Level 6

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2014-13 Level 6

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

82 is a composite number. 82 = 1 x 82 or  2 x 41. Factors of 82: 1, 2, 41, 82. Prime factorization: 82 = 2 x 41.

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

2014-13 Level 6 Logic

81 and Level 5

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2014-13 Level 5

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

81 is a composite number and is 9 squared. 81 = 1 x 81, 3 x 27, or 9 x 9. Factors of 81: 1, 3, 9, 27, 81. Prime factorization: 81 = 3 x 3 x 3 x 3 which can also be written 3⁴.

Since √81 = 9, a whole number, 81 is a perfect square.

Sometimes 81 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact we use is 9 x 9 = 81.

2014-13 Level 5 Logic