Level 3 Puzzle

320 Christmas Factor Trees

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  • 320 is a composite number.
  • Prime factorization: 320 = 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 320 = 2⁶ x 5
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 x 2 = 14. Therefore 320 has exactly 14 factors.
  • Factors of 320: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 160, 320
  • Factor pairs: 320 = 1 x 320, 2 x 160, 4 x 80, 5 x 64, 8 x 40, 10 x 32, or 16 x 20
  • Taking the factor pair with the largest square number factor, we get √320 = (√5)(√64) = 8√5 ≈ 17.889

In these factor trees for 320, we can also see factor trees for 4, 8, 10, 16, 20, 32, 40, 64, 80, and 160. The tops of each of those factor trees are in brown. No matter which factor pairs we use to build the tree, we always get the same prime factors which are indicated in red.

320 Factor Trees

Now try finding the factors in this Christmas tree puzzle that make it work like a multiplication table:

2014-49 Level 3

Print the puzzles or type the factors on this excel file: 10 Factors 2014-12-08

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.

2014-49 Level 3 Factors

 

312 and Virgács

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Today might not be the 312th day of the year, but here’s a fun fact about the number 312:

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  • 312 is a composite number.
  • Prime factorization: 312 = 2 x 2 x 2 x 3 x 13, which can be written 312 = (2^3) x 3 x 13
  • 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 312 has exactly 16 factors.
  • Factors of 312: 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312
  • Factor pairs: 312 = 1 x 312, 2 x 156, 3 x 104, 4 x 78, 6 x 52, 8 x 39, 12 x 26, or 13 x 24
  • Taking the factor pair with the largest square number factor, we get √312 = (√4)(√78) = 2√78 ≈ 17.664

The night of December 5th, Mikulás, or St. Nickolas, will visit homes in Hungary, some neighboring countries, and even a few houses in the United States.  Children who have been good will awaken to find their shoes filled with little treats such as candy, fruit or nuts. Since all children are occasionally a little bit naughty, they will also find virgács, a small collection of twigs that have been spray-painted gold and decoratively bound together.  Virgács are not sold in the United States, so St Nickolas will be making some himself using the bristles from a natural broom.  The finished product should look like today’s puzzle.

This puzzle is meant to be reminiscent of the virgács my children’s ancestors probably received each year on St Nickolas Day.
2014-48 Level 3
Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-01
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.
2014-48 Level 3 Factors

305 and Level 3

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  • 305 is a composite number.
  • Prime factorization: 305 = 5 x 61
  • 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 305 has exactly 4 factors.
  • Factors of 305: 1, 5, 61, 305
  • Factor pairs: 305 = 1 x 305 or 5 x 61
  • 305 has no square factors that allow its square root to be simplified. √305 ≈ 17.464

2014-47 Level 3

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

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.

2014-47 Level 3 Factors

298 and Level 3

/ ivasallay / 1 Comment

  • 298 is a composite number.
  • Prime factorization: 298 = 2 x 149
  • 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 298 has exactly 4 factors.
  • Factors of 298: 1, 2, 149, 298
  • Factor pairs: 298 = 1 x 298 or 2 x 149
  • 298 has no square factors that allow its square root to be simplified. √298 ≈ 17.263

Can you fill out this 1 – 12 multiplication table when you’re only given twelve clues?

2014-46 Level 3

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

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.

2014-46 Level 3 Factors

292 and Level 3

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  • 292 is a composite number.
  • Prime factorization: 292 = 2 x 2 x 73, which can be written 292 = 2² x 73
  • 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 292 has exactly 6 factors.
  • Factors of 292: 1, 2, 4, 73, 146, 292
  • Factor pairs: 292 = 1 x 292, 2 x 146, or 4 x 73
  • Taking the factor pair with the largest square number factor, we get √292 = (√4)(√73) = 2√73 ≈ 17.088.

Can you fill out this multiplication table that has only ten clues?

2014-45 Level 3

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-10

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.

2014-45 Level 3 Factors

286 and Level 3

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  • 286 is a composite number.
  • Prime factorization: 286 = 2 x 11 x 13
  • 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 286 has 8 factors.
  • Factors of 286: 1, 2, 11, 13, 22, 26, 143, 286
  • Factor pairs: 1 x 286, 2 x 143, 11 x  26, or 13 x 22
  • 286 has no square factors that allow its square root to be simplified. √286 ≈ 16.912

2014-44 Level 3

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-03

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.

2014-44 Level 3 Factors

278 and Level 3

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  • 278 is a composite number.
  • Prime factorization: 278 = 2 x 139
  • 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 278 has 4 factors.
  • Factors of 278: 1, 2, 139, 278
  • Factor pairs: 278 = 1 x 278 or 2 x 139
  • 278 has no square factors that allow its square root to be simplified. √278 ≈ 16.673

Here is today’s puzzle:

2014-43 Level 3

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-27

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.

2014-43 Level 3 Factors

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

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

255 and Level 3

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  • 255 is a composite number.
  • Prime factorization: 255 = 3 x 5 x 17
  • 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 255 has 8 factors.
  • Factors of 255: 1, 3, 5, 15, 17, 51, 85, 255
  • Factor pairs: 255 = 1 x 255, 3 x 85, 5 x 51, or 15 x 17
  • 255 has no square factors that allow its square root to be simplified. √255 ≈ 15.969

Here’s a factoring puzzle for you to solve:

2014-40 Level 3

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

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.

2014-40 Level 3 Factors