Square root

271 and Level 3

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  • 271 is a prime number.
  • Prime factorization: 271 is prime.
  • The exponent of prime number 271 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 271 has exactly 2 factors.
  • Factors of 271: 1, 271
  • Factor pairs: 271 = 1 x 271
  • 271 has no square factors that allow its square root to be simplified. √271 ≈ 16.462

How do we know that 271 is a prime number? If 271 were not a prime number, then it would be divisible by at least one prime number less than or equal to √271 ≈ 16.462. Since 271 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 271 is a prime number.

Here is today’s factoring puzzle:

2014-42 Level 3

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

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.

2014-42 Level 3 Factors

270 and Level 2

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Today’s Puzzle:

Can you solve this level 2 puzzle?

2014-42 Level 2

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

Factors of 270:

  • 270 is a composite number.
  • Prime factorization: 270 = 2 x 3 x 3 x 3 x 5, which can be written 2 x (3^3) x 5
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 270 has 16 factors.
  • Factors of 270: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270
  • Factor pairs: 270 = 1 x 270, 2 x 135, 3 x 90, 5 x 54, 6 x 45, 9 x 30, 10 x 27, or 15 x 18
  • Taking the factor pair with the largest square number factor, we get √270 = (√9)(√30) = 3√30 ≈ 16.432

Sum-Difference Puzzles:

30 has four factor pairs. One of those factor pairs adds up to 13, and another subtracts to 13. Can you find those factor pairs to solve the first puzzle below?

270 has eight factor pairs. One of them adds up to 39, and a different one subtracts to 39. 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 would I say that?

Factors for Today’s Puzzle:

2014-42 Level 2 Factors

 

269 and Five More Consecutive Square Roots

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  • 269 is a prime number.
  • Prime factorization: 269 is prime.
  • The exponent of prime number 269 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 269 has exactly 2 factors.
  • Factors of 269: 1, 269
  • Factor pairs: 269 = 1 x 269
  • 269 has no square factors that allow its square root to be simplified. √269 ≈ 16.401

How do we know that 269 is a prime number? If 269 were not a prime number, then it would be divisible by at least one prime number less than or equal to √269 ≈ 16.401. Since 269 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 269 is a prime number.

As I have previously written, 844, 845, 846, 847, and 848 are the smallest FIVE consecutive numbers whose square roots can be simplified. Here are the second smallest FIVE with the same property.

1680 square roots

The first number in the second set, 1680, equals 2 x 840 which is very close to the first number in the first set. Will strings of five consecutive numbers with reducible square roots occur about once every 850 numbers?

We can find the number of factors for these numbers by examining their prime factorizations.

1680 prime factorization

The number of factors for each of the integers in this second set ranges from 3 to 40. Only two of the integers have the same number of factors. Finding another string of four or more numbers that have reducible square roots as well as the same number of factors may be difficult.

268 and Level 1

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  • 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

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  • 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

265 and Level 5

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  • 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

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  • 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

263 How and Why You Should Show Your Work

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  • 263 is a prime number.
  • Prime factorization: 263 is prime.
  • The exponent of prime number 263 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 263 has exactly 2 factors.
  • Factors of 263: 1, 263
  • Factor pairs: 263 = 1 x 263
  • 263 has no square factors that allow its square root to be simplified. √263 ≈ 16.217

How do we know that 263 is a prime number? If 263 were not a prime number, then it would be divisible by at least one prime number less than or equal to √263 ≈ 16.217. Since 263 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 263 is a prime number.

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Joseph Nebus reads hundreds of comics every day as he looks for ones with a mathematical theme. He regularly shares these finds on his blog and writes a short explanation about the mathematics mentioned in the comics. I especially loved his reading-the-comics-october-14-2014 edition.

Why mathematics students should show their work is clearly explained under Jeff Mallet’s Frazz (October 12) comic strip.  Joseph Nebus basically gives two reasons to show work. The first reason has probably been stated by teachers thousands of times, but the second is truly an inspiration, and I highly recommend teachers and students alike read it!

How should the work be shown? This You-tube video does a very good job showing how to show work and make that work as readable as possible.

262 and Level 3

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  • 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