422 has three even digits. How many factors could it possibly have? Scroll down to see.

This Find the Factors puzzle has the same color scheme that puzzles 414 and 417 had, but this one also has a couple of numbers that aren’t colored at all.  If 422 were colored, it would be yellow. I’ve had one very close guess posted in the comments for 417. Will anybody be able to read my mind and figure out what the coloring is all about? You can type your guess in the comments. Only an elementary education is required to figure it out!

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

• 422 is a composite number.
• Prime factorization: 422 = 2 x 211
• 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 422 has exactly 4 factors.
• Factors of 422: 1, 2, 211, 422
• Factor pairs: 422 = 1 x 422 or 2 x 211
• 422 has no square factors that allow its square root to be simplified. √422 ≈ 20.5426

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.

405 times 2 equals 810, and 810 divided by 10 is 81. And guess what, 405 divided by 5 is also 81. Multiplying by .2 instead of dividing by 5 makes a nice division trick!

All of 405’s factors are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

• 405 is a composite number.
• Prime factorization: 405 = 3 x 3 x 3 x 3 x 5, which can be written 405 = (3^4) x 5
• The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 405 has exactly 10 factors.
• Factors of 405: 1, 3, 5, 9, 15, 27, 45, 81, 135, 405
• Factor pairs: 405 = 1 x 405, 3 x 135, 5 x 81, 9 x 45, or 15 x 27
• Taking the factor pair with the largest square number factor, we get √405 = (√81)(√5) = 9√5 ≈ 20.1246

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.

88 is divisible by 4 so 388 is also divisible by 4 and is a composite number. 88 is also divisible by 8, but the third to the last digit is odd, making 388 NOT divisible by 8.

Print the puzzles or type the factors on this excel file:12 Factors 2015-02-09

• 388 is a composite number.
• Prime factorization: 388 = 2 x 2 x 97, which can be written 388 = (2^2) x 97
• 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 388 has exactly 6 factors.
• Factors of 388: 1, 2, 4, 97, 194, 388
• Factor pairs: 388 = 1 x 388, 2 x 194, or 4 x 97
• Taking the factor pair with the largest square number factor, we get √388 = (√4)(√97) = 2√97 ≈ 19.698

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.

380 is even so it has to be a composite number. Scroll down past the puzzle for all of its factors.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

• 380 is a composite number.
• Prime factorization: 380 = 2 x 2 x 5 x 19, which can be written 380 = (2^2) x 5 x 19
• 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 380 has exactly 12 factors.
• Factors of 380: 1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380
• Factor pairs: 380 = 1 x 380, 2 x 190, 4 x 95, 5 x 76, 10 x 38, or 19 x 20
• Taking the factor pair with the largest square number factor, we get √380 = (√4)(√95) = 2√95 ≈ 19.494

Since those consecutive numbers in the last factor pair are the last possibility on the chart, we know that 380 is the sum of the first 19 even numbers. Thus,

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 = 380. Those numbers can be rearranged this way:

2 + 36 + 4 + 34 + 6 + 32 + 8 + 30 + 10 + 28 + 12 + 26 + 14 + 24 + 16 + 22 + 18 + 20 + 38 = 380, which is more easily seen as the sum of ten 38’s.

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.

362 is even so it is a composite number. Its factors are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-19

• 362 is a composite number.
• Prime factorization: 362 = 2 x 181
• 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 362 has exactly 4 factors.
• Factors of 362: 1, 2, 181, 362
• Factor pairs: 362 = 1 x 362 or 2 x 181
• 362 has no square factors that allow its square root to be simplified. √362 ≈ 19.026

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.

When 2^353 is divided by 353, the remainder is 2, so 353 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

• 353 is a prime number.
• Prime factorization: 353 is prime.
• The exponent of prime number 353 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 353 has exactly 2 factors.
• Factors of 353: 1, 353
• Factor pairs: 353 = 1 x 353
• 353 has no square factors that allow its square root to be simplified. √353 ≈ 18.788

How do we know that 353 is a prime number? If 353 were not a prime number, then it would be divisible by at least one prime number less than or equal to √353 ≈ 18.788. Since 353 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 353 is a prime number.

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.