356 is even so it is a composite number. Scroll down past the puzzle to see its factors.

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

• 356 is a composite number.
• Prime factorization: 356 = 2 x 2 x 89, which can be written 356 = (2^2) x 89
• 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 356 has exactly 6 factors.
• Factors of 356: 1, 2, 4, 89, 178, 356
• Factor pairs: 356 = 1 x 356, 2 x 178, or 4 x 89
• Taking the factor pair with the largest square number factor, we get √356 = (√4)(√89) = 2√89 ≈ 18.868

Five can evenly divide into numbers when the last digit is 5, so 355 is a composite number. Scroll down past the puzzle to see its factors.

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

• 355 is a composite number.
• Prime factorization: 355 = 5 x 71
• 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 355 has exactly 4 factors.
• Factors of 355: 1, 5, 71, 355
• Factor pairs: 355 = 1 x 355 or 5 x 71
• 355 has no square factors that allow its square root to be simplified. √355 ≈ 18.841

354 is even and thus divisible by 2, so 354 is a composite number. It is also made from three consecutive numbers so it is divisible by 3. Scroll down past the puzzle to see its factors.

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

• 354 is a composite number.
• Prime factorization: 354 = 2 x 3 x 59
• 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 354 has exactly 8 factors.
• Factors of 354: 1, 2, 3, 6, 59, 118, 177, 354
• Factor pairs: 354 = 1 x 354, 2 x 177, 3 x 118, or 6 x 59
• 354 has no square factors that allow its square root to be simplified. √354 ≈ 18.815

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.

350 is even and thus divisible by 2, so 350 is a composite number. Scroll down past the puzzle to see its factors.

In pose-a-puzzle Ed Southall (solvemymaths) showed examples and described several great mathematical puzzles that are available on the internet. He wrote, “Not only do they excite students in a way that perhaps no textbook page can, they also make you think. I don’t think any good puzzles can be done just by applying a mindless algorithm to a familiar looking structure.”

It will take a while, but check out all the different types of puzzles listed. Here’s something he wrote specifically about the Find the Factors puzzles: “They are the best way I’ve found to get students of *any* ability to practice multiplication tables.”

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

• 350 is a composite number.
• Prime factorization: 350 = 2 x 5 x 5 x 7, which can be written 350 = 2·5²·7
• The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 350 has exactly 12 factors.
• Factors of 350: 1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350
• Factor pairs: 350 = 1 x 350, 2 x 175, 5 x 70, 7 x 50, 10 x 35, or 14 x 25
• Taking the factor pair with the largest square number factor, we get √350 = (√14)(√25) = 5√14 ≈ 18.708