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

2014-47 Level 6

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

2014-47 Level 6 Logic

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307 and Level 5

  • 307 is a prime number.
  • Prime factorization: 307 is prime.
  • 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:

2014-47 Level 5

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

2014-47 Level 5 Logic

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 3² 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!

2014-47 Level 4

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)

2014-47 Level 4 Logic

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

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

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?

2014-47 Level 2

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

2014-47 Level 2 Factors

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

2014-47 Level 1

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

2014-47 Level 1 Factors

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.

2014-46 Level 6

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.

Assuming 6 goes over 72 in top row

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

2014-46 Level 6 Logic

 

301 and Level 5

  • 301 is a composite number.
  • Prime factorization: 301 = 7 x 43
  • 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 301 has exactly 4 factors.
  • Factors of 301: 1, 7, 43, 301
  • Factor pairs: 301 = 1 x 301 or 7 x 43
  • 301 has no square factors that allow its square root to be simplified. √301 ≈ 17.349

2014-46 Level 5

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

2014-46 Level 5 Logic

Integers (up to 300) with the Same Number of Factors

  • 300 is a composite number.
  • Prime factorization: 300 = 2 x 2 x 3 x 5 x 5, which can be written 300 = (2^2) x 3 x (5^2)
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 x 2 x 3 = 18. Therefore 300 has exactly 18 factors.
  • Factors of 300: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300
  • Factor pairs: 300 = 1 x 300, 2 x 150, 3 x 100, 4 x 75, 5 x 60, 6 x 50, 10 x 30, 12 x 25 or 15 x 20
  • Taking the factor pair with the largest square number factor, we get √300 = (√3)(√100) = 10√3 ≈ 17.321

I made a couple of graphics a few weeks ago anticipating my 300th post.

Integers with the Same Number of Factors

Observations:

  • The only number with exactly one factor is one.
  • The number of factors of all the black integers on the chart are powers of 2. These integers have irreducible square roots.
  • Consecutive integers with the same number of factors can only occur if the number of factors is NOT a prime number.
  • 90/300 or 30% of the first 300 integers have 4 factors. (Largest group)
  • 62/300 or 20.6666% of the first 300 integers are prime numbers and therefore have 2 factors. (2nd largest group)
  • 49/300 or 16.3333% of the first 300 integers have 8 factors. (3rd largest group)
  • 40/300 or 13.3333% of the first 300 integers have 6 factors. (4th largest group) All integers with 6 factors have reducible square roots.
  • 22/300 or 7.3333% of the first 300 integers have 12 factors. (5th largest group)
  • All but 37/300 or 12.3333% of the first 300 integers are in one of those 5 groups.
  • 118/300 or 39.3333% of the first 300 integers have square roots that can be simplified.

Warning: the next chart and observations could make your brain hurt:

Smallest Numbers with 2 to 20 Factors

  • How many numbers have exactly 2 factors? Euclid proved that there is an infinite number of prime numbers which means there is an infinite number of integers with exactly 2 factors.
  • How many integers have exactly 19 factors? Even though the smallest integer with exactly 19 factors is 262,144, there is still an infinite number of integers with exactly that many factors. The integers in that list are each prime number raised to the 18th power. Since there is an infinite number of prime numbers, there is an infinite number of integers with exactly 19 factors.
  • {2⁹⁹⁶, 3⁹⁹⁶, 5⁹⁹⁶, . . . } is the infinite list of integers with exactly 997 factors. Likewise {2ᵖ⁻¹, 3ᵖ⁻¹, 5ᵖ⁻¹, . . . } where p is a prime number is the infinite list of integers with exactly p factors.
  • If the number of factors is c, a composite number, then it could be said that there is more than an infinite number of integers with that many factors because the infinite list of integers will include {2ᶜ⁻¹, 3ᶜ⁻¹, 5ᶜ⁻¹, . . . } as well as many other integers.

299 and Level 4

  • 299 is a composite number.
  • Prime factorization: 299 = 13 x 23
  • 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 299 has exactly 4 factors.
  • Factors of 299: 1, 13, 23, 299
  • Factor pairs: 299 = 1 x 299 or 13 x 23
  • 299 has no square factors that allow its square root to be simplified. √299 ≈ 17.292

Warning: The logic needed to solve this Level 4 puzzle is complicated.

2014-46 Level 4

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

2014-46 Level 4 Logic