• 220 is a composite number.
• Prime factorization: 220 = 2 x 2 x 5 x 11, which can be written (2^2) x 5 x 11
• The exponents in the prime factorization are 2, 1 and 1. Adding one to each and multiplying we get (2 +1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 220 has 12 factors.
• Factors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220
• Factor pairs: 220 = 1 x 220, 2 x 110, 4 x 55, 5 x 44, 10 x 22,  or 11 x 20
• Taking the factor pair with the largest square number factor, we get √220 = (√4)( √55) = 2√55 ≈ 14.832

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-09-01

• 219 is a composite number.
• Prime factorization: 219 = 3 x 73
• 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 219 has 4 factors.
• Factors of 219: 1, 3, 73, 219
• Factor pairs: 219 = 1 x 219 or 3 x 73
• 219 has no square factors so its square root cannot be simplified. √219 ≈ 14.799

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-09-01

• 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

• 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

Today’s Puzzle:

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

Steps That Use Logic to Solve That Puzzle:

Factors of 216:

• 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

Sum-Difference Puzzle:

54 has four factor pairs. One of those factor pairs adds up to 15, and another one subtracts to 15. Those factor pairs will help you solve the first puzzle below.

216 has eight factor pairs. One of them adds up to 30 and a different one subtracts to 30. If you can identify those factor pairs, then you can solve the second puzzle.

The second puzzle is really just the first puzzle in disguise. Why did I say that?

 

• 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 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

• 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