• 238 is a composite number.
• Prime factorization: 238 = 2 x 7 x 17
• The exponents in the prime factorization are 1, 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 238 has 8 factors.
• Factors of 238: 1, 2, 7, 14, 17, 34, 119, 238
• Factor pairs: 238 = 1 x 238, 2 x 119, 7 x 34, or 14 x 17
• Since 238 has no square number factors, its square root cannot be simplified. √238 ≈ 15.427

Print the puzzles or type the factors on this excel file: 12 Factors 2014-09-22

• 237 is a composite number.
• Prime factorization: 237 = 3 x 79
• 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 237 has 4 factors.
• Factors of 237: 1, 3, 79, 237
• Factor pairs: 237 = 1 x 237 or 3 x 79
• Since 237 has no square number factors, its square root cannot be simplified. √237 ≈ 15.395

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-15

• 236 is a composite number.
• Prime factorization: 236 = 2 x 2 x 59, which can be written (2^2) x 59
• 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 236 has 6 factors.
• Factors of 236: 1, 2, 4, 59, 118, 236
• Factor pairs: 236 = 1 x 236, 2 x 118, or 4 x 59
• Taking the factor pair with the largest square number factor, we get √236 = (√4)( √59) = 2√59 ≈ 15.362

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-15

• 235 is a composite number.
• Prime factorization: 235 = 5 x 47
• 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 235 has 4 factors.
• Factors of 235: 1, 5, 47, 235
• Factor pairs: 235 = 1 x 235 or 5 x 47
• Since 235 has no square number factors, its square root cannot be simplified. √235 ≈ 15.3297

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-15

• 234 is a composite number.
• Prime factorization: 234 = 2 x 3 x 3 x 13, which can be written 2 x (3^2) x 13
• The exponents in the prime factorization are 1, 2 and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 234 has 12 factors.
• Factors of 234: 1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234
• Factor pairs: 234 = 1 x 234, 2 x 117, 3 x 78, 6 x 39, 9 x 26, or 13 x 18
• Taking the factor pair with the largest square number factor, we get √234 = (√9)( √26) = 3√26 ≈ 15.297

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-15

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.

• 232 is a composite number.
• Prime factorization: 232 = 2 x 2 x 2 x 29, which can be written (2^3) x 29
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 232 has 8 factors.
• Factors of 232: 1, 2, 4, 8, 29, 58, 116, 232
• Factor pairs: 232 = 1 x 232, 2 x 116, 4 x 58, or 8 x 29
• Taking the factor pair with the largest square number factor, we get √232 = (√4)( √58) = 2√58 ≈ 15.232

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-15

• 231 is a composite number and a triangular number.
• Prime factorization: 231 = 3 x 7 x 11
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 231 has 8 factors.
• Factors of 231: 1, 3, 7, 11, 21, 33, 77, 231
• Factor pairs: 231 = 1 x 231, 3 x 77, 7 x 33, or 11 x 21
• 231 has no square factors so its square root cannot be simplified. √231 ≈ 15.199

• Because 21 + 1 = 22 and 21 x (22/2) = 231, we know 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 = 231

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-09-08

• 230 is a composite number.
• Prime factorization: 230 = 2 x 5 x 23
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 230 has 8 factors.
• Factors of 230: 1, 2, 5, 10, 23, 46, 115, 230
• Factor pairs: 230 = 1 x 230, 2 x 115, 5 x 46, or 10 x 23
• 230 has no square factors so its square root cannot be simplified. √230 ≈ 15.166

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-09-08