## A Multiplication Based Logic Puzzle

### 308 and Level 6

• 308 is a composite number.
• Prime factorization: 308 = 2 × 2 × 7 × 11, which can be written 308 = 2² × 7 × 11
• 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 308 has exactly 12 factors.
• Factors of 308: 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308
• Factor pairs: 308 = 1 x 308, 2 x 154, 4 x 77, 7 x 44, 11 x 28, or 14 x 22
• Taking the factor pair with the largest square number factor, we get √308 = (√4)(√77) = 2√77 ≈ 17.5499

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

### 307 and Level 5

• 307 is a prime number.
• Prime factorization: 307 is prime and cannot be factored.
• The exponent of prime number 307 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 307 has exactly 2 factors.
• Factors of 307: 1, 307
• Factor pairs: 307 = 1 x 307
• 307 has no square factors that allow its square root to be simplified. √307 ≈ 17.521

How do we know that 307 is a prime number? If 307 were not a prime number, then it would be divisible by at least one prime number less than or equal to √307 ≈ 17.521. Since 307 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 307 is a prime number.

Eleven clues that might stump you:

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

### 306 and Level 4

• 306 is a composite number.
• Prime factorization: 306 = 2 x 3 x 3 x 17, which can be written 306 = 2 x (3^2) x 17
• 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 306 has exactly 12 factors.
• Factors of 306: 1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306
• Factor pairs: 306 = 1 x 306, 2 x 153, 3 x 102, 6 x 51, 9 x 34, or 17 x 18
• Taking the factor pair with the largest square number factor, we get √306 = (√9)(√34) = 3√34 ≈ 17.493

Happy Thanksgiving!

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

Here’s a little more about the number 306:

306 = 17 × 18, which means it is the sum of the first 17 even numbers.

• Thus, 2 + 4 + 6 + 8 + . . .  + 30 + 32 + 34 = 306

It also means that we are halfway between 17² and 18², or halfway between 289 and 324. The average of those two numbers is 306.5.

AND it means that 17² + 18² – 1 = 2(306) = 2(17 × 18)

### 305 and Level 3

• 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

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.

### 304 and Level 2

• 304 is a composite number.
• Prime factorization: 304 = 2 x 2 x 2 x 2 x 19, which can be written 304 = (2^4) x 19
• 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 304 has exactly 10 factors.
• Factors of 304: 1, 2, 4, 8, 16, 19, 38, 76, 152, 304
• Factor pairs: 304 = 1 x 304, 2 x 152, 4 x 76, 8 x 38, or 16 x 19
• Taking the factor pair with the largest square number factor, we get √304 = (√16)(√19) = 4√19 ≈ 17.436

This 1 – 10 multiplication table has only twelve clues. Can you still complete the entire table?

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

### 303 and Level 1

• 303 is a composite number.
• Prime factorization: 303 = 3 x 101
• 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 303 has exactly 4 factors.
• Factors of 303: 1, 3, 101, 303
• Factor pairs: 303 = 1 x 303 or 3 x 101
• 303 has no square factors that allow its square root to be simplified. √303 ≈ 17.407

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

### 302 and Level 6

• 302 is a composite number.
• Prime factorization: 302 = 2 x 151
• 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 302 has exactly 4 factors.
• Factors of 302: 1, 2, 151, 302
• Factor pairs: 302 = 1 x 302 or 2 x 151
• 302 has no square factors that allow its square root to be simplified. √302 ≈ 17.378

Here’s a very tricky level 6 puzzle to try.

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

The logic for this one is complicated. Here is an explanation that may be helpful:

9 is a clue two times in this puzzle. At most one of those 9’s can be 3 x 3. Clue 99 and the other nine will use both 9’s so 72 cannot be 8 x 9 in this puzzle. Here is an explanation of why 72 must be 6 x 12 with the 6 in the first column and the 12 in the top row.

Now we can find the rest of the factors easily using logic:
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