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.

We can easily see that 421 cannot be evenly divided by 2, 3, or 5. Could it possibly be a prime number? Scroll down to know for sure.

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

• 421 is a prime number. 419 and 421 are twin primes.
• Prime factorization: 421 is prime.
• The exponent of prime number 421 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 421 has exactly 2 factors.
• Factors of 421: 1, 421
• Factor pairs: 421 = 1 x 421
• 421 has no square factors that allow its square root to be simplified. √421 ≈ 20.5183

How do we know that 421 is a prime number? If 421 were not a prime number, then it would be divisible by at least one prime number less than or equal to √421 ≈ 20.5183. Since 421 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 421 is a prime number.

2^419 (mod 419) = 2 so 419 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 2015-03-09

• 419 is a prime number.
• Prime factorization: 419 is prime.
• The exponent of prime number 419 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 419 has exactly 2 factors.
• Factors of 419: 1, 419
• Factor pairs: 419 = 1 x 419
• 419 has no square factors that allow its square root to be simplified. √419 ≈ 20.4695

How do we know that 419 is a prime number? If 419 were not a prime number, then it would be divisible by at least one prime number less than or equal to √419 ≈ 20.4695. Since 419 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 419 is a prime number.

4 + 1 + 7 = 12, a multiple of three, so 417 can be evenly divided by three. Its other factors are listed below the puzzle.

The colors for today’s puzzle use the same key that puzzle 414 used and 422 will use, even though none of the squares here are blue. If I had colored the actual number 417, it would be blue. Also if I colored the number 414 it would have been green.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

• 417 is a composite number.
• Prime factorization: 417 = 3 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 417 has exactly 4 factors.
• Factors of 417: 1, 3, 139, 417
• Factor pairs: 417 = 1 x 417 or 3 x 139
• 417 has no square factors that allow its square root to be simplified. √417 ≈ 20.4206

415 is obviously a multiple of 5 so it is a composite number. Scroll down past the puzzle to view its other factors.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

• 415 is a composite number.
• Prime factorization: 415 = 5 x 83
• 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 415 has exactly 4 factors.
• Factors of 415: 1, 5, 83, 415
• Factor pairs: 415 = 1 x 415 or 5 x 83
• 415 has no square factors that allow its square root to be simplified. √415 ≈ 20.3715

413 can’t be divided evenly by 2, 3, or 5. Can it be divided by 7? Here’s a couple of quick divisibility tricks to try, each requiring the last number to be separated from the rest giving us 41 and 3.

1. Double the 3 and subtract it from 41. That gives us 41 – 6 = 35, a multiple of 7 so 413 can be divided evenly by 7.
2. Multiply the 3 by five and add it to 41. That gives us 41 + 15 = 56, a multiple of 7 as well, so we can draw the same conclusion.

Thank you to abyssbrain for telling me about the second divisibility trick. All of 413’s factors are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

• 413 is a composite number.
• Prime factorization: 413 = 7 x 59
• 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 413 has exactly 4 factors.
• Factors of 413: 1, 7, 59, 413
• Factor pairs: 413 = 1 x 413 or 7 x 59
• 413 has no square factors that allow its square root to be simplified. √413 ≈ 20.3224

Because 12 is divisible by 4, we know that 412 is also divisible by 4.

The number 12 has six factors. Does 412 have more than that? Find out after the puzzle!

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

• 412 is a composite number.
• Prime factorization: 412 = 2 x 2 x 103, which can be written 412 = (2^2) x 103
• 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 412 has exactly 6 factors.
• Factors of 412: 1, 2, 4, 103, 206, 412
• Factor pairs: 412 = 1 x 412, 2 x 206, or 4 x 103
• Taking the factor pair with the largest square number factor, we get √412 = (√4)(√103) = 2√103 ≈ 20.2978

Surprised?