number puzzle

176 and Level 3

/ ivasallay / Leave a comment
176 is a composite number.
Factor pairs: 176 = 1 x 176, 2 x 88, 4 x 44, 8 x 22, or 11 x 16
Factors of 176: 1, 2, 4, 8, 11, 16, 22, 44, 88, 176
Prime factorization: 176 = 2 x 2 x 2 x 2 x 11 which can be written (2^4) x 11.

2014-28 Level 3

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-07-14

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 (1st column) and factor row (top 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 factor column and the factor row one cell at a time as you go.

2014-28 Level 3 Factors

175 and Level 2

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175 is a composite number.
Factor pairs: 175 = 1 x 175, 5 x 35, or 7 x 25
Factors of 175, 1, 5, 7, 25, 35, 175
Prime factorization: 175 = 5 x 5 x 7 which can be written 175 = (5^2) x 7

When the double-the-last-digit-and-subtract-it-from-the-remaining-numbers divisibility trick for 7 is applied to 175 we get: 17 – (5 x 2) = 17 – 10 = 7, a multiple of 7, so 175 can be evenly divided by 7.

2014-28 Level 2

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-07-14

2014-28 Level 2 Factors

 

174 and Level 1

/ ivasallay / Leave a comment
174 is a composite number.
Factor pairs: 174 = 1 x 174, 2 x 87, 3 x 58, 6 x 29
Factors of 174: 1, 2, 3, 6, 29, 58, 87, 174
Prime factorization: 174 = 2 x 3 x 29

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 2014-28 Level 1

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-07-14

2014-28 Level 1 Factors

173 and Level 6

/ ivasallay / Leave a comment
  • 173 is a prime number.
  • Prime factorization: 173 is prime.
  • The exponent of prime number 173 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 173 has exactly 2 factors.
  • Factors of 173: 1, 173
  • Factor pairs: 173 = 1 x 173
  • 173 has no square factors that allow its square root to be simplified. √173 ≈ 13.1529

How do we know that 173 is a prime number? If 173 were not a prime number, then it would be divisible by at least one prime number less than or equal to √173 ≈ 13.2. Since 173 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 173 is a prime number.

2014-27 Level 6

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07

2014-27 Level 6 Logic

172 and Level 5

/ ivasallay / Leave a comment
172 is a composite number.
Factor pairs: 172 = 1 x 172, 2 x 86, or 4 x 43
Factors of 172: 1, 2, 4, 43, 86, 172
Prime factorization: 172 = 2 x 2 x 43, which can be written (2^2) x 43

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2014-27 Level 5

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07

2014-27 Level 5 Logic

171 and Level 4

/ ivasallay / Leave a comment
171 is a composite number.
Factor pairs: 171 = 1 x 171, 3 x 57, or 9 x 19
Factors of 171: 1, 3, 9, 19, 57, 171
Prime factorization: 170 = 3 x 3 x 19, which can be written 170 = (3^2) x 19

171 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 and is therefore a triangular number.

The fact that 171 is a triangular number is easy to recognize because 170 = 9 x 19 = (18/2) x 19. Any integer resulting from n(n + 1)/2 is a triangular number.

2014-27 Level 4

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07

2014-27 Level 4 Logic

170 and Level 3

/ ivasallay / Leave a comment
170 is a composite number.
Factor pairs: 170 = 1 x 170, 2 x 85, 5 x 34, or 10 x 17
Factors of 170: 1, 2, 5, 10, 17, 34, 85, 170
Prime factorization: 170 = 2 x 5 x 17

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2014-27 Level 3

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07

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 (1st column) and factor row (top 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 factor column and the factor row one cell at a time as you go.

2014-27 Level 3 Factors

169 and Level 2

/ ivasallay / Leave a comment
169 is a composite number and a perfect square.
Factor pairs: 169 = 1 x 169 or 13 x 13
Factors of 169: 1, 13, 169
Prime factorization: 169 = 13 x 13 which can be written 13²

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2014-27 Level 2

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07

2014-27 Level 2 Factors

167 and Level 1

/ ivasallay / Leave a comment
  • 167 is a prime number.
  • Prime factorization: 167 is prime.
  • The exponent of prime number 167 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 167 has exactly 2 factors.
  • Factors of 167: 1, 167
  • Factor pairs: 167 = 1 x 167
  • 167 has no square factors that allow its square root to be simplified. √167 ≈ 12.9228

How do we know that 167 is a prime number? If 167 were not a prime number, then it would be divisible by at least one prime number less than or equal to √167 ≈ 12.9. Since 167 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 167 is a prime number.

2014-27 Level 1

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07

2014-27 Level 1 Factors

166 and Level 6

/ ivasallay / 2 Comments
166 is a composite number.
Factor pairs: 166 = 1 x 166 or 2 x 83
Factors of 166: 1, 2, 83, 166
Prime factorization: 166 = 2 x 83

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2014-26 Level 6

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-30

2014-26 Level 6 Logic