218 and Level 6

• 218 is a composite number.
• Prime factorization: 218 = 2 x 109
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1) x (1 + 1) = 2 x 2 = 4. Therefore 218 has 4 factors.
• Factors of 218: 1, 2, 109, 218
• Factor pairs: 218 = 1 x 218 or 2 x 109
• 218 has no square factors so its square root cannot be simplified. √218 ≈ 14.765

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-25

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217 and Level 5

• 217 is a composite number.
• Prime factorization: 217 = 7 x 31
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 217 has 4 factors.
• Factors of 217: 1, 7, 31, 217
• Factor pairs: 217 = 1 x 217 or 7 x 31
• Since 217 has no square factors, its square root cannot be simplified. √217 ≈ 14.731

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-25

216 and Level 4

• 216 is a composite number.
• 216 = 6 x 6 x 6, so it is a perfect cube, 6³.
• Prime factorization: 216 = 2 x 2 x 2 x 3 x 3 x 3, which can be written 216 = 2³ x 3³.
• The exponents in the prime factorization are 3 and 3. Adding one to each and multiplying we get (3 + 1) x (3 + 1) = 4 x 4 = 16. Therefore 216 has 16 factors.
• Factors of 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
• Factor pairs: 216 = 1 x 216, 2 x 108, 3 x 72, 4 x 54, 6 x 36, 8 x 27, 9 x 24, 12 x 18
• Taking the factor pair with the largest square number factor, we get √216 = (√6)(√36) = 6√6 ≈ 14.697

It’s been a busy summer for me and lots of other people, yet Alan Parr found the time this summer to write a great post about the Find the Factors puzzles.  You can read it at established1962 find-the-factors.

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-25

215 and Level 3

• 215 is a composite number.
• Prime factorization: 215 = 5 x 43
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1) x (1 + 1) = 2 x 2 = 4. Therefore 215 has 4 factors.
• Factors of 215: 1, 5, 43, 215
• Factor pairs: 215 = 1 x 215 or 5 x 43
• Since 215 has no square factors, the square root of 215 cannot be simplified. √215 ≈ 14.663

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-25

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.

214 and Level 2

• 214 is a composite number.
• Prime factorization: 214 = 2 x 107
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1) x (1 + 1) = 2 x 2 = 4. Therefore 214 has 4 factors.
• Factors of 214: 1, 2, 107, 214
• Factor pairs: 214 = 1 x 214 or 2 x 107
• Since 214 has no square factors, the square root of 214 cannot be simplified. √214 ≈ 14.629

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-25

213 and Level 1

• 213 is a composite number.
• Prime factorization: 213 = 3 x 71
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 213 has 4 factors.
• Factors of 213: 1, 3, 71, 213
• Factor pairs: 213 = 1 x 213 or 3 x 71
• Since 213 has no square factors, the square root of 213 cannot be simplified. √213 ≈ 14.595

I recommend reading a-math-book-to-change-your-teaching, an article written by a fellow blogger about a book that simplifies the teaching of mathematics while making the subject more meaningful for students.

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-25

212 and Level 6

• 212 is a composite number.
• Prime factorization: 212 = 2 x 2 x 53, which can be written (2^2) x 53
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 212 has 6 factors.
• Factors of 212: 1, 2, 4, 53, 106, 212
• Factor pairs: 212 = 1 x 212, 2 x 106, or 4 x 53
• Taking the factor pair with the largest square number factor, we get √212 = (√4)(√53) = 2√58 ≈ 14.5602

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-18

211 and Level 5

• 211 is a prime number.
• Prime factorization: 211 is prime.
• The exponent of prime number 211 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 211 has exactly 2 factors.
• Factors of 211: 1, 211
• Factor pairs: 211 = 1 x 211
• 211 has no square factors so its square root cannot be simplified. √211 ≈ 14.526

How do we know that 211 is a prime number? If 211 were not a prime number, then it would be divisible by at least one prime number less than or equal to √211 ≈ 14.526. Since 211 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 211 is a prime number.

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-18

210 and Level 4

• 210 is a composite number and a triangular number.
• Prime factorization: 210 = 2 x 3 x 5 x 7, so 210 is the product of the first four prime numbers!
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 210 has 16 factors.
• Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
• Factor pairs: 210 = 1 x 210, 2 x 105, 3 x 70, 5 x 42, 6 x 35, 7 x 30, 10 x 21, or 14 x 15
• Since 210 has no square factors, √210 cannot be simplified. √210 ≈ 14.491

Because 20 + 1 = 21 and (20/2) x 21 = 210, we know that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 210, thus it is the 20th triangular number.

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-18

209 and Level 3

2-0+9=11 so 209 is divisible by 11.

• 209 is a composite number.
• Prime factorization: 209 = 11 x 19
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 209 has exactly 4 factors.
• Factors of 209: 1, 11, 19, 209
• Factor pairs: 209 = 1 x 209 or 11 x 19
• 209 has no square factors that allow its square root to be simplified. √209 ≈ 14.456832

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Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-18

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.