factor

180 and Level 1

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Today’s Puzzle:

2014-29 Level 1
Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-21

Factors of 180:

180 is a composite number.
Factor pairs: 180 = 1 x 180, 2 x 90, 3 x 60, 4 x 45, 5 x 36, 6 x 30, 9 x 20, 10 x 18, or 12 x 15
Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
Prime factorization: 180 = 2 x 2 x 3 x 3 x 5 which can be written 180 = (2^2) x (3^2) x 5

Sum-Difference Puzzle:

180 has nine factor pairs. One of the factor pairs adds up to 41, and another one subtracts to 41. If you can identify those two factor pairs, then you can solve this puzzle!

More about the Number 180:

There are 180 degrees in a triangle, and a straight angle is 180 degrees.

180 is the hypotenuse of a Pythagorean triple:
108-144-180, which is (3-4-5) times 36.

Solution to This Week’s Puzzle:

2014-29 Level 1 Factors

179 and Level 6

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  • 179 is a prime number.
  • Prime factorization: 179 is prime.
  • The exponent of prime number 179 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 179 has exactly 2 factors.
  • Factors of 179: 1, 179
  • Factor pairs: 179 = 1 x 179
  • 179 has no square factors that allow its square root to be simplified. √179 ≈ 13.379088

How do we know that 179 is a prime number? If 179 were not a prime number, then it would be divisible by at least one prime number less than or equal to √179 ≈ 13.4. Since 179 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 179 is a prime number.

2014-28 Level 6

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

2014-28 Level 6 Logic

178 and Level 5

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178 is a composite number.
Factor pairs: 178 = 1 x 178 or 2 x 89
Factors of 178: 1, 2, 89, 178
Prime factorization: 178 = 2 x 89

2014-28 Level 5

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

 2014-28 Level 5 Logic

177 and Level 4

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177 is a composite number.
Factor pairs: 177 = 1 x 177, 3 x 59
Factors of 177: 1, 3, 59, 177
Prime factorization: 177 = 3 x 59

Here is a magic square made entirely of prime numbers:

prime magic square

What number do you get when you add up the numbers of any one of its rows, columns, or diagonals?

According to Wikipedia, you can’t get a smaller magic sum than 177 when all the numbers in the magic square are prime numbers. Wolfram MathWorld has several Prime Magic Squares of various dimensions up to 13 × 13. Two of the squares include the number 1, which is not a prime number but is sometimes permitted to be in a prime magic square because it is not a composite number.

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2014-28 Level 4

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

2014-28 Level 4 Logic

176 and Level 3

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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

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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

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  • 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

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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

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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