Multiplication table

340 and Level 5

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  • 340 is a composite number.
  • Prime factorization: 340 = 2 x 2 x 5 x 17, which can be written 340 = (2^2) x 5 x 17
  • 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 340 has exactly 12 factors.
  • Factors of 340: 1, 2, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340
  • Factor pairs: 340 = 1 x 340, 2 x 170, 4 x 85, 5 x 68, 10 x 34, or 17 x 20
  • Taking the factor pair with the largest square number factor, we get √340 = (√4)(√85) = 2√85 ≈ 18.439

Do you have any goals to improve your thinking power this year? Well, solving puzzles like this one can be good for your brain!

2014-52 Level 5

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 5 Logic

339 and Level 4

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  • 339 is a composite number.
  • Prime factorization: 339 = 3 x 113
  • 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 339 has exactly 4 factors.
  • Factors of 339: 1, 3, 113, 339
  • Factor pairs: 339 = 1 x 339 or 3 x 113
  • 339 has no square factors that allow its square root to be simplified. √339 ≈ 18.412

Existential Spaghetti wrote a blog post a couple of years ago on New Year’s resolutions being yearly-chinese-finger-traps. People tend to choose resolutions that are “both time-consuming and often high-energy.” Ponder that as you solve this Level 4 Chinese Finger Trap Factoring Puzzle made especially for the New Year.

2014-52 Level 4

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

 Amazon.com sells these Chinese Finger Traps:

2014-52 Level 4 Logic

337 What Will Be the Factors of 2015?

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Since this is my 337th post, I’ll first give some information about the number 337, then I will predict the factors for 2015.

  • 337 is a prime number.
  • Prime factorization: 337 is prime.
  • The exponent of prime number 337 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 337 has exactly 2 factors.
  • Factors of 337: 1, 337
  • Factor pairs: 337 = 1 x 337
  • 337 has no square factors that allow its square root to be simplified. √337 ≈ 18.358

How do we know that 337 is a prime number? If 337 were not a prime number, then it would be divisible by at least one prime number less than or equal to √337 ≈ 18.358. Since 337 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 337 is a prime number.

Celebrate the New Year by giving this puzzle a try!

2014-52 Level 3

WHAT will be the FACTORS of the YEAR 2015?

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 3 Factors

On New Year’s Eve 2013 I predicted that the positive factors for 2014 would be 1, 2, 19, 38, 53, 106,1007, and 2014, and my predictions were 100% accurate!

On this last day of 2014, I boldly announce my predictions for the factors of the year 2015:

  • The positive factors for 2015 will be 1, 5, 13, 31, 65, 155, 403, and 2015
  • Some of these factors will occur in pairs: 1 and 2015, 5 and 403, 13 and 155, as well as 31 and 65.
  • Unfortunately there will be some negative factors in 2015 as well. They will be -1, -5, -13, -31, -65, -155, -403, and -2015.

Whatever life throws your way, I wish you a happy, healthy, and prosperous 2015.

336 and Level 2

/ ivasallay / 2 Comments

Today’s Puzzle:

The sixteen clues given in this puzzle are all you need to complete this multiplication table!

2014-52 Level 2

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 2 Factors

Factors of 336:

  • 336 is a composite number.
  • Prime factorization: 336 = 2 x 2 x 2 x 2 x 3 x 7, which can be written 336 = (2^4) x 3 x 7
  • 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 336 has exactly 20 factors.
  • Factors of 336: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336
  • Factor pairs: 336 = 1 x 336, 2 x 168, 3 x 112, 4 x 84, 6 x 56, 7 x 48, 8 x 42, 12 x 28, 14 x 24 or 16 x 21
  • Taking the factor pair with the largest square number factor, we get √336 = (√16)(√21) = 4√21 ≈ 18.330

Sum-Difference Puzzles:

84 has six factor pairs. One of those factor pairs adds up to 25, and another one subtracts to 25. Can you determine what those factor pairs are to solve the first puzzle below?

336 has ten factor pairs. One of them adds up to 50, and a different one subtracts to 50. 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?

 

335 and Level 1

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  • 335 is a composite number.
  • Prime factorization: 335 = 5 x 67
  • 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 335 has exactly 4 factors.
  • Factors of 335: 1, 5, 67, 335
  • Factor pairs: 335 = 1 x 335 or 5 x 67
  • 335 has no square factors that allow its square root to be simplified. √335 ≈ 18.303

If you can multiply, divide and use a little logic, you should easily be able to complete this multiplication table puzzle.

2014-52 Level 1

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 1 Factors

334 and Level 6

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  • 334 is a composite number.
  • Prime factorization: 334 = 2 x 167
  • 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 334 has exactly 4 factors.
  • Factors of 334: 1, 2, 167, 334
  • Factor pairs: 334 = 1 x 334 or 2 x 167
  • 334 has no square factors that allow its square root to be simplified. √334 ≈ 18.276

Solving this Level 6 puzzle can be a little tricky: Is 1 or 2 the common factor for 6 and 8? Is 3 or 6 the common factor for 12 and 30? Is 4 or 8 the common factor for 40 and 16? In each case only one of those choices will make this puzzle work as a multiplication table? Can you figure out what those choices should be? Use logic to find the correct solution, not trial and error.

2014-51 Level 6

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

2014-51 Level 6 Logic

333 and Level 5

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  • 333 is a composite number.
  • Prime factorization: 333 = 3 x 3 x 37, which can be written 333 = 3² x 37
  • 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 333 has exactly 6 factors.
  • Factors of 333: 1, 3, 9, 37, 111, 333
  • Factor pairs: 333 = 1 x 333, 3 x 111, or 9 x 37
  • Taking the factor pair with the largest square number factor, we get √333 = (√9)(√37) = 3√37 ≈ 18.248

Can the eleven clues in this puzzle help you to complete this multiplication table?

2014-51 Level 5

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

2014-51 Level 5 Logic

326 Tiny Christmas Factor Tree

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  • 326 is a composite number.
  • Prime factorization: 326 = 2 x 163
  • 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 326 has exactly 4 factors.
  • Factors of 326: 1, 2, 163, 326
  • Factor pairs: 326 = 1 x 326 or 2 x 163
  • 326 has no square factors that allow its square root to be simplified. √326 ≈ 18.055

Even though 326 is a three digit number, there is only one way to construct its factor tree, two if you count its mirror image. Either way is illustrated here. Below them is a Christmas factor tree puzzle that is a lot more interesting than the factor tree for 326.

2014-50 Level 4

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-15

2014-50 Level 4 Logic

325 is a Triangular Number

/ ivasallay / 3 Comments

  • 325 is a composite number.
  • Prime factorization: 325 = 5 x 5 x 13, which can be written 325 = 5² x 13
  • 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 325 has exactly 6 factors.
  • Factors of 325: 1, 5, 13, 25, 65, 325
  • Factor pairs: 325 = 1 x 325, 5 x 65, or 13 x 25
  • Taking the factor pair with the largest square number factor, we get √325 = (√13)(√25) = 5√13 ≈ 18.028

325 is a triangular number. 1 + 2 + 3 + 4 + . . . + 22 + 23 + 24 + 25 = 325. Shorthand for that sum of 25 numbers is given with the ∑ sign in the graphic below. The way to find the sum quickly using multiplication is also given:

325 - Triangular Number

If you look at this Level 3 puzzle from far enough away, you also might see a triangle!

2014-50 Level 3

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-15

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.

2014-50 Level 3 Factors

324 Christmas Factor Trees

/ ivasallay / 2 Comments

  • 324 is a composite number.
  • Prime factorization: 324 = 2 x 2 x 3 x 3 x 3 x 3, which can be written 324 = (2^2) x (3^4)
  • The exponents in the prime factorization are 2 and 4. Adding one to each and multiplying we get (2 + 1)(4 + 1) = 3 x 5 = 15. Therefore 324 has exactly 15 factors.
  • Factors of 324: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324
  • Factor pairs: 324 = 1 x 324, 2 x 162, 3 x 108, 4 x 81, 6 x 54, 9 x 36, 12 x 27, or 18 x 18
  • 324 is a perfect square. √324 = 18

Included in these 324 factor trees are smaller factor trees for the numbers in brown: 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, and 162. Prime factors of 324 are in red.

324 Factor Trees

This Level 2 puzzle looks like part of a Christmas tree branch. It’s not too difficult to solve. Give it a try!

2014-50 Level 2

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-15

2014-50 Level 2 Factors