Composite number

268 and Level 1

/ ivasallay / Leave a comment

  • 268 is a composite number.
  • Prime factorization: 268 = 2 x 2 x 67, which can be written (2^2) x 67
  • 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 268 has 6 factors.
  • Factors of 268: 1, 2, 4, 67, 134, 268
  • Factor pairs: 268 = 1 x 268, 2 x 134, or 4 x 67
  • Taking the factor pair with the largest square number factor, we get √268 = (√4)(√67) = 2√67 ≈ 16.371.

Here is today’s puzzle:

2014-42 Level 1

Print the puzzles or type the factors on this excel file: 12 Factors 2014-10-20

2014-42 Level 1 Factors

267 and Level 6

/ ivasallay / Leave a comment

  • 267 is a composite number.
  • Prime factorization: 267 = 3 x 89
  • 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 267 has 4 factors.
  • Factors of 267: 1, 3, 89, 267
  • Factor pairs: 267 = 1 x 267 or 3 x 89
  • 267 has no square factors that allow its square root to be simplified. √267 ≈ 16.3401

Can you solve this level 6 puzzle? Possible steps to logically arrive at the solution are given at the end of the post.

2014-41 Level 6

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

2014-41 Level 6 Logic

266 Why 8 Consecutive Numbers with 6 Factors is Impossible

/ ivasallay / 1 Comment

  • 266 is a composite number.
  • Prime factorization: 266 = 2 x 7 x 19
  • 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 266 has 8 factors.
  • Factors of 266: 1, 2, 7, 14, 19, 38, 133, 266
  • Factor pairs: 265 = 1 x 266, 2 x 133, 7 x 38, or 14 x 19
  • 266 has no square factors that allow its square root to be simplified. √266 ≈ 16.3095

242, 243, 244, and 245 are the smallest four consecutive numbers that have the same number of factors. Each of them has exactly six factors, and as a result each one of their square roots can be simplified. Is it possible to have a longer string of consecutive numbers with exactly six factors? I don’t know yet. It seems reasonable that it could happen, so I am on the lookout for five, six, or seven consecutive numbers that have exactly six factors.

I won’t bother looking for a string of eight or more numbers with six factors. I already know that would be impossible. Here’s an example illustrating why:

844 - 848 prime factorization

846 ruins the run because it has an additional prime factor that doubles the number of factors that 846 has in all. Further down the number line that problem might be overcome in a different set of consecutive numbers.

However, the problem with 848 is a recurring problem that will never be overcome. 848 is divisible by 8, as is every eighth number. The prime factorization of numbers that are divisible by eight must contain a power of two that is greater than or equal to three. Its number of factors calculation would have to be at least (3 + 1)(1 + 1) = 4 x 2 = 8. (The ONLY number divisible by 8 that has exactly 6 factors is 32.)

Even though the numbers from 844 to 848 don’t have the same number of factors, they still have a distinction. They are the smallest five consecutive numbers whose square roots can be simplified!

844, 845, 846, 847, 848

265 and Level 5

/ ivasallay / Leave a comment

  • 265 is a composite number.
  • Prime factorization: 265 = 5 x 53
  • 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 265 has 4 factors.
  • Factors of 265: 1, 5, 53, 265
  • Factor pairs: 265 = 1 x 265 or 5 x 53
  • 265 has no square factors that allow its square root to be simplified. √265 ≈ 16.279

Here is today’s puzzle:

2014-41 Level 5

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

2014-41 Level 5 Logic

264 and Level 4

/ ivasallay / Leave a comment

  • 264 is a composite number.
  • Prime factorization: 264 = 2 x 2 x 2 x 3 x 11, which can be written (2^3) x 3 x 11
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 264 has 16 factors.
  • Factors of 264: 1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264
  • Factor pairs: 264 = 1 x 264, 2 x 132, 3 x 88, 4 x 66, 6 x 44, 8 x 33, 11 x 24, or 12 x 22
  • Taking the factor pair with the largest square number factor, we get √264 = (√4)(√66) = 2√66 ≈ 16.248

That’s a lot of factors. Finding the factors for this puzzle will be much easier:

2014-41 Level 4

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

2014-41 Level 4 Logic

262 and Level 3

/ ivasallay / Leave a comment

  • 262 is a composite number.
  • Prime factorization: 262 = 2 x 131
  • 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 262 has 4 factors.
  • Factors of 262: 1, 2, 131, 262
  • Factor pairs: 262 = 1 x 262 or 2 x 131
  • 262 has no square factors that allow its square root to be simplified. √262 ≈ 16.186

Can you solve this factoring puzzle?

2014-41 Level 3

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

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.

2014-41 Level 3 Factors

261 and Level 2

/ ivasallay / Leave a comment

  • 261 is a composite number.
  • Prime factorization: 261 = 3 x 3 x 29
  • 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 261 has 6 factors.
  • Factors of 261: 1, 3, 9, 29, 87, 261
  • Factor pairs: 261 = 1 x 261, 3 x 87, 9 x 29
  • Taking the factor pair with the largest square number factor, we get √261 = (√9)(√29) = 3√29 ≈ 16.155

This puzzle will look like a multiplication table after you find the factors and then fill in the rest of the table.

2014-41 Level 2

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

2014-41 Level 2 Factors

260 Some Thoughts on Those Four Consecutive Numbers

/ ivasallay / 3 Comments

  • 260 is a composite number.
  • Prime factorization: 260 = 2 x 2 x 5 x 13, which can be written 260 = (2^2) x 5 x 13
  • 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 260 has 12 factors.
  • Factors of 260: 1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260
  • Factor pairs: 260 = 1 x 260, 2 x 130, 4 x 65, 5 x 52, 10 x 26, or 13 x 20
  • Taking the factor pair with the largest square number factor, we get √260 = (√4)(√65) = 2√65 ≈ 16.125

Recently I wrote about the smallest four-consecutive-numbers whose square roots could all be simplified. The same numbers were also the smallest four consecutive numbers to have the same number of factors.

4 numbers; 6 factors

Each of those numbers had 6 factors, and guess what, ANY number with exactly 6 factors can have its square root simplified. The prime factorization of ANY number with exactly 6 factors can be expressed in one of the three following ways:

6 factors

Since numbers with six factors always have a prime factor raised to a power greater than one, they can always have their square roots simplified. The fact that those four consecutive numbers have the same number of factors makes them extraordinary; that they all can have their square roots simplified is merely the natural consequence of that extraordinary fact.

259 and Level 1

/ ivasallay / Leave a comment

  • 259 is a composite number.
  • Prime factorization: 259 = 7 x 37
  • 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 259 has 4 factors.
  • Factors of 259: 1, 7, 37, 259
  • Factor pairs: 259 = 1 x 259 or 7 x 37
  • 259 has no square factors that allow its square root to be simplified. √259 ≈ 16.093

Here’s a level 1 Find the Factors puzzle:

2014-41 Level 1

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

2014-41 Level 1 Factors

258 and Level 6

/ ivasallay / Leave a comment

  • 258 is a composite number.
  • Prime factorization: 258 = 2 x 3 x 43
  • 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 258 has 8 factors.
  • Factors of 258: 1, 2, 3, 6, 43, 86, 129, 258
  • Factor pairs: 258 = 1 x 258, 2 x 129, 3 x 86, or 6 x 43
  • 258 has no square factors that allow its square root to be simplified. √258 ≈ 16.062

Level 6 puzzles aren’t easy. Can you solve this one?

2014-40 Level 6

Print the puzzles or type the factors on this excel file: 12 Factors 2014-10-06

2014-40 Level 6 Logic