• 305 is a composite number.
• Prime factorization: 305 = 5 x 61
• 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 305 has exactly 4 factors.
• Factors of 305: 1, 5, 61, 305
• Factor pairs: 305 = 1 x 305 or 5 x 61
• 305 has no square factors that allow its square root to be simplified. √305 ≈ 17.464

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

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.

• 304 is a composite number.
• Prime factorization: 304 = 2 x 2 x 2 x 2 x 19, which can be written 304 = (2^4) x 19
• The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 304 has exactly 10 factors.
• Factors of 304: 1, 2, 4, 8, 16, 19, 38, 76, 152, 304
• Factor pairs: 304 = 1 x 304, 2 x 152, 4 x 76, 8 x 38, or 16 x 19
• Taking the factor pair with the largest square number factor, we get √304 = (√16)(√19) = 4√19 ≈ 17.436

This 1 – 10 multiplication table has only twelve clues. Can you still complete the entire table?

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

• 303 is a composite number.
• Prime factorization: 303 = 3 x 101
• 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 303 has exactly 4 factors.
• Factors of 303: 1, 3, 101, 303
• Factor pairs: 303 = 1 x 303 or 3 x 101
• 303 has no square factors that allow its square root to be simplified. √303 ≈ 17.407

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

• 302 is a composite number.
• Prime factorization: 302 = 2 x 151
• 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 302 has exactly 4 factors.
• Factors of 302: 1, 2, 151, 302
• Factor pairs: 302 = 1 x 302 or 2 x 151
• 302 has no square factors that allow its square root to be simplified. √302 ≈ 17.378

Here’s a very tricky level 6 puzzle to try.

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

The logic for this one is complicated. Here is an explanation that may be helpful:

9 is a clue two times in this puzzle. At most one of those 9’s can be 3 x 3. Clue 99 and the other nine will use both 9’s so 72 cannot be 8 x 9 in this puzzle. Here is an explanation of why 72 must be 6 x 12 with the 6 in the first column and the 12 in the top row.

Now we can find the rest of the factors easily using logic:

• 301 is a composite number.
• Prime factorization: 301 = 7 x 43
• 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 301 has exactly 4 factors.
• Factors of 301: 1, 7, 43, 301
• Factor pairs: 301 = 1 x 301 or 7 x 43
• 301 has no square factors that allow its square root to be simplified. √301 ≈ 17.349

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

• 299 is a composite number.
• Prime factorization: 299 = 13 x 23
• 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 299 has exactly 4 factors.
• Factors of 299: 1, 13, 23, 299
• Factor pairs: 299 = 1 x 299 or 13 x 23
• 299 has no square factors that allow its square root to be simplified. √299 ≈ 17.292

Warning: The logic needed to solve this Level 4 puzzle is complicated.

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

• 298 is a composite number.
• Prime factorization: 298 = 2 x 149
• 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 298 has exactly 4 factors.
• Factors of 298: 1, 2, 149, 298
• Factor pairs: 298 = 1 x 298 or 2 x 149
• 298 has no square factors that allow its square root to be simplified. √298 ≈ 17.263

Can you fill out this 1 – 12 multiplication table when you’re only given twelve clues?

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

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.

• 297 is a composite number.
• Prime factorization: 297 = 3 x 3 x 3 x 11 which can be written 297 = 3³ x 11
• 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 297 has exactly 8 factors.
• Factors of 297: 1, 3, 9, 11, 27, 33, 99, 297
• Factor pairs: 297 = 1 x 297, 3 x 99, 9 x 33, or 11 x 27
• Taking the factor pair with the largest square number factor, we get √297 = (√9)(√33) = 3√33 ≈ 17.234

This multiplication table has only 14 clues. Is that enough to complete it?

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

• 296 is a composite number.
• Prime factorization: 296 = 2 x 2 x 2 x 37 which can be written 296 = 2³ x 37
• 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 296 has exactly 8 factors.
• Factors of 296: 1, 2, 4, 8, 37, 74, 148, 296
• Factor pairs: 296 = 1 x 296, 2 x 148, 4 x 74, or 8 x 37
• Taking the factor pair with the largest square number factor, we get √296 = (√4)(√74) = 2√74 ≈ 17.2046505

Although I came home from the hospital on Sunday, I haven’t felt like spending much time on the computer reading or writing anything. Full recovery will be another five weeks, but hopefully I’ll start feeling better in less time.  I do feel better today than I did on Sunday.

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

• 295 is a composite number.
• Prime factorization: 295 = 5 x 59
• 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 295 has exactly 4 factors.
• Factors of 295: 1, 5, 59, 295
• Factor pairs: 295 = 1 x 295 or 5 x 59
• 295 has no square factors that allow its square root to be simplified. √295 ≈ 17.176

These eleven clues could be pretty tricky!

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-10