A Multiplication Based Logic Puzzle

Archive for September, 2014

246 and Level 1

  • 246 is a composite number.
  • Prime factorization: 246 = 2 x 3 x 41
  • 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 246 has 8 factors.
  • Factors of 246: 1, 2, 3, 6, 41, 82, 123, 246
  • Factor pairs: 246 = 1 x 246, 2 x 123, 3 x 82, or 6 x 41
  • 246 has no square factors so its square root cannot be simplified. √246 ≈ 15.684

 2014-39 Level 1

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-29

2014-39 Level 1 Factors

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245 – The Last of Four Consecutive Numbers

  • 245 is a composite number.
  • Prime factorization: 245 = 5 x 7 x 7, which can be written 245 = 5 x (7^2)
  • The exponents in the prime factorization are 1 and 2. Adding one  to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3 = 6. Therefore 245 has 6 factors.
  • Factors of 245: 1, 5, 7, 35, 49, 245
  • Factor pairs: 245 = 1 x 245, 5 x 49, or 7 x 35
  • Taking the factor pair with the largest square number factor, we get √245 = (√5)(√49) = 7√5 ≈ 15.652

Square roots 242 - 245

 

I was surprised when I noticed that the square roots of these 4 consecutive numbers – 242, 243, 244, and 245 could all be simplified.

The square root of a whole number can only be simplified if that whole number has a square number as one of its factors. All four of these numbers meet that condition, and they are the first four consecutive numbers to do so.

For numbers less than or equal to 240, there are only 3 sets of 3 consecutive square roots that can be simplified.

  • √48 = 4√3
  • √49 = 7
  • √50 = 5√2
  • √98 = 7√2
  • √99 = 3√11
  • √100 = 10
  • √124 = 2√31
  • √125 = 5√5
  • √126 = 3√14

242, 243, 244, and 245 also have another distinction. They each have exactly 6 factors and are the smallest consecutive four numbers to have the same number of factors.

244 and Level 6

  • 244 is a composite number.
  • Prime factorization: 244 = 2 x 2 x 61, which can be written 244 = (2^2) x 61
  • 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 244 has 6 factors.
  • Factors of 244: 1, 2, 4, 61, 122, 244
  • Factor pairs: 244 = 1 x 244, 2 x 122, 4 x 61
  • Taking the factor pair with the largest square number factor, we get √244 = (√4)(√61) = 2√61 ≈ 15.6205

2014-38 Level 6

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

2014-38 Level 6 Logic

243 and Level 5

  • 243 is a composite number.
  • Prime factorization: 243 = 3 x 3 x 3 x 3 x 3, which can be written 243 = (3^5)
  • The exponent in the prime factorization is 5. Adding one we get (5 + 1) = 6. Therefore 243 has 6 factors.
  • Factors of 243: 1, 3, 9, 27, 81, 243
  • Factor pairs: 243 = 1 x 243, 3 x 81, or 9 x 27
  • Taking the factor pair with the largest square number factor, we get √243 = (√3)(√81) = 9√3 ≈ 15.588

Happy birthday, Sue

This puzzle is for my sister, who is great at completing these Level 5 puzzles. Here is a black and white version for those who may prefer it:

2014-38 Level 5

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

2014-38 Level 5 Logic

242 and Level 4

  • 242 is a composite number.
  • Prime factorization: 242 = 2 x 11 x 11, which can be written 2 x (11^2)
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3 = 6. Therefore 242 has 6 factors.
  • Factors of 242: 1, 2, 11, 22, 121, 242
  • Factor pairs: 242 = 1 x 242, 2 x 121, or 11 x 22
  • Taking the factor pair with the largest square number factor, we get √242 = (√2)(√121) = 11√2 ≈ 15.556

2014-38 Level 4

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

2014-38 Level 4 Logic

241 and Level 3

  • 241 is a prime number.
  • Prime factorization: 241 is prime.
  • The exponent of prime number 241 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 241 has exactly 2 factors.
  • Factors of 241: 1, 241
  • Factor pairs: 241 = 1 x 241
  • 241 has no square factors so its square root cannot be simplified. √241 ≈ 15.524

How do we know that 241 is a prime number? If 241 were not a prime number, then it would be divisible by at least one prime number less than or equal to √241 ≈ 15.524. Since 241 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 241 is a prime number.

2014-38 Level 3

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

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.

2014-38 Level 3 Factors

A Forest of 240 Factor Trees

  • 240 is a composite number.
  • Prime factorization: 240 = 2 x 2 x 2 x 2 x 3 x 5, which can be written 2⁴ x 3 x 5
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 240 has 20 factors.
  • Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
  • Factor pairs: 240 = 1 x 240, 2 x 120, 3 x 80, 4 x 60, 5 x 48, 6 x 40, 8 x 30, 10 x 24, 12 x 20, or 15 x 16
  • Taking the factor pair with the largest square number factor, we get √240 = (√16)(√15) = 4√15 ≈ 15.492

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 = 240; that’s the sum of the first 15 even numbers.

Because 240 has so many factors, it is possible to make MANY different factor trees that create a forest of 240 factor trees. This post only contains eleven of those many possibilities. The two trees below demonstrate different permutations that can be made from the same basic tree. The mirror images of both, as well as mirror images of parts of either tree, would be other permutations.

240 Factor Trees

 

A good way to make a factor tree for a composite number is to begin with one of its factor pairs and then make factor trees for the composite numbers in that factor pair.

240 Factor Trees 1 - 3

In this first set of three factor trees we can also see the factor trees for 120, 80, 4, & 60.

240 Factor Trees 4 - 6

These three factor trees also include factor trees for 48, 6, 40, 8, and 30.

240 Factor Trees 7 - 9

Finally, these three factor trees also include factor trees for 10, 24, 12, 20, 15, and 16.

This forest of 240 factor trees is dedicated to Joseph Nebus. Read the comments to his post, You might also like, because, I don’t know why, to discover why I was inspired to create images of parts of this forest.

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