A Multiplication Based Logic Puzzle

Archive for August, 2015

How Many Factors Do the Numbers Up to 600 Have?

600 ties 360, 420, 480, 504, and 540 for having more factors than all other previous whole numbers.

600 is the sum of consecutive prime numbers 293 and 307.

600 is the hypotenuse of two Pythagorean triples: 360-480-600 and 168-576-600. What are the greatest common factors of each of those triples?

In my 300th, 400th, or 500th posts, I reviewed the amount of factors numbers up to 500 have. I even had a horse race for the numbers from 401 to 500 because there were three lead changes which made watching a gif of it a little more interesting.

For the numbers from 501 to 600, there is no horse race: 4 was the first out of the gate and soon left all the other factor amounts in the dust. Click on the graphic below to see it better.

501 to 600 Same Number of Factors

I also like to keep track of how many integers have reducible square roots. 40% of these numbers do, but the total numbers from 1 to 600 that have reducible square roots is a little lower than that as you can see in the chart below that also shows the amount of numbers up to 600 which have a particular number of factors:

Total Number of Integers with the Same Amounts of Factors 1 - 600

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  • 600 is a composite number.
  • Prime factorization: 600 = 2 x 2 x 2 x 3 x 5 x 5, which can be written 600 = (2^3) x 3 x (5^2)
  • The exponents in the prime factorization are 3, 1 and 2. Adding one to each and multiplying we get (3 + 1)(1 + 1)(2 + 1) = 4 x 2 x 3 = 24. Therefore 600 has exactly 24 factors.
  • Factors of 600: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600
  • Factor pairs: 600 = 1 x 600, 2 x 300, 3 x 200, 4 x 150, 5 x 120, 6 x 100, 8 x 75, 10 x 60, 12 x 50, 15 x 40, 20 x 30 or 24 x 25
  • Taking the factor pair with the largest square number factor, we get √600 = (√100)(√6) = 10√6 ≈ 24.494897. (Note: 2 + 4 = 6 and 49 + 48 = 97 so this square root and √6 can be easy to remember.)

 

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What often happens to a number like 599 when a number next to it has so many factors?

The square root of any number from 576 to 624 is between 24 and 25. That means the first number in any of their factor pairs will be 24 or less. The combined number of factor pairs for the 49 integers from 576 to 624 is 189. The number of factor pairs for any given number ranges from 1 to 12. Let’s look at the averages: The mean (189/49) is 3.857 factor pairs per number. The median is 3 factor pairs, and the mode (the number of factor pairs that occurs most often) is 4 factor pairs.

Distribution of the Number of Factor Pairs from 576 to 624

What happens to a number like 599 when a number next to it has far more than the average number of factor pairs? Quite often, but not always, that number has no choice but to be a prime number.

Number of Factor Pairs around 599

Even though 600 has only three prime factors (2, 3 and 5), it still managed to be divisible by 50% of the numbers from 1 to 24, and there just aren’t many possibilities left for the numbers immediately before or after it.

The twelve numbers less than or equal to 24 that will divide into 600 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20 and 24.

Because 9, 14, 16, 18, 21, and 22 each have 2 or 3 as a prime factor, those six numbers also are not possible factors of 599 or 601.

Every whole number is divisible by 1, but besides that, there are only six numbers available as possible factors for those two numbers: 7, 11, 13, 17, 19, and 23. Since neither 599 nor 601 is divisible by any of those numbers, they turn out to be twin primes.

Usually at least one of the numbers before or after a number with far more than its fair share of factor pairs will be a prime number.

119 and 121, the numbers before and after 120 are notable exceptions. √120 ≈ 10.95441. The factors of 120 that are less than or equal to 10 are 1, 2, 3, 4, 5, 6, 8, and 10 which is 80% of the possible factors. Yet 119 managed to be divisible by 7, and 121 managed to be divisible by 11 so neither one of them is a prime number.

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If you add up the digits of 499, 589, or 598, you will get 22.

599 is the smallest whole number whose digits add up to 23. Thank you, Stetson.edu for that number fact.

  • 599 is a prime number and a twin prime with 601.
  • Prime factorization: 599 is prime.
  • The exponent of prime number 599 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 599 has exactly 2 factors.
  • Factors of 599: 1, 599
  • Factor pairs: 599 = 1 x 599
  • 599 has no square factors that allow its square root to be simplified. √599 ≈ 24.4744765

How do we know that 599 is a prime number? If 599 were not a prime number, then it would be divisible by at least one prime number less than or equal to √599 ≈ 24.5. Since 599 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 599 is a prime number.

598 and Level 6

598 is the sum of all the prime number from 17 to 71.

598 is also the hypotenuse of the Pythagorean triple 230-552-598. What is the greatest common factor of those three numbers?

Here’s another easy as 1-2-3 number fact from Stetson.edu: (5^1) + (9^2) +(8^3) = 598.

Since 23 + 3 = 26 and 23 x 26 = 598, we know we are only 2 away from 24 x 25 = 600.

598 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 598 is a composite number.
  • Prime factorization: 598 = 2 x 13 x 23
  • 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 598 has exactly 8 factors.
  • Factors of 598: 1, 2, 13, 23, 26, 46, 299, 598
  • Factor pairs: 598 = 1 x 598, 2 x 299, 13 x 46, or 23 x 26
  • 598 has no square factors that allow its square root to be simplified. √598 ≈ 24.4540385.

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598 Logic

597 and Level 5

597 is made from three consecutive odd numbers so it is divisible by 3.

Because it is divisible by 3, it is the sum of three consecutive numbers. It is also the sum of three consecutive odd numbers:

  • 198 + 199 + 200 = 597
  • 197 + 199 + 201 = 597

Why was 199 the middle number in both of those sums?

Stetson.edu informs us that even though (597!!! + 1) generates an error message on our calculators, it is indeed a very large prime number. Whoever discovered that fact must have been extremely excited.

597 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 597 is a composite number.
  • Prime factorization: 597 = 3 x 199
  • 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 597 has exactly 4 factors.
  • Factors of 597: 1, 3, 199, 597
  • Factor pairs: 597 = 1 x 597 or 3 x 199
  • 597 has no square factors that allow its square root to be simplified. √597 ≈ 24.43358

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597 Logic

596 Select the Correct Top Row

596 is the sum of the four primes numbers from 139 to 157. The two prime numbers in the middle of that sum are twin primes. Do you know what they are?

596 is also the hypotenuse of the Pythagorean triple 204-560-596. What is the greatest common factor of those three numbers?

596 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 596 is a composite number.
  • Prime factorization: 596 = 2 x 2 x 149, which can be written 596 = (2^2) x 149
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 596 has exactly 6 factors.
  • Factors of 596: 1, 2, 4, 149, 298, 596
  • Factor pairs: 596 = 1 x 596, 2 x 298, or 4 x 149
  • Taking the factor pair with the largest square number factor, we get √596 = (√4)(√149) = 2√149 ≈ 24.41311123

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596 Logic

595 and Level 3

595 = (34/2)(34 + 1) so 595 is the 34th triangular number.

Thus 1 + 2 + 3 + 4 + . . . . . + 31 + 32 + 33 + 34 = 595

595 is also the hypotenuse of four Pythagorean triples. Each of these triples has its own greatest common factor.

  • 91-588-595
  • 252-539-595
  • 280-525-595
  • 357-476-595

595 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 595 is a composite number.
  • Prime factorization: 595 = 5 x 7 x 17
  • 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 595 has exactly 8 factors.
  • Factors of 595: 1, 5, 7, 17, 35, 85, 119, 595
  • Factor pairs: 595 = 1 x 595, 5 x 119, 7 x 85, or 17 x 35
  • 595 has no square factors that allow its square root to be simplified. √595 ≈ 24.39262.

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A Logical Approach to solve a FIND THE FACTORS puzzle: 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.

595 Factors

 

594 and Level 2

It is as easy as 1-2-3 to remember this number fact from Stetson.edu:

(1^5) + (2^9) + (3^4) = 594

594 is the sum of the ten prime numbers from 41 to 79.

594 is even so it is divisible by 2.

5 + 9 + 4 = 18 so 594 is divisible by both 3 and 9 (and by 6 and 18 because 594 is even)

5 – 9 + 4 = 0 so 594 is divisible by 11.

594 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 594 is a composite number.
  • Prime factorization: 594 = 2 x 3 x 3 x 3 x 11, which can be written 594 = 2 x (3^3) x 11
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 594 has exactly 16 factors.
  • Factors of 594: 1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 99, 198, 297, 594
  • Factor pairs: 594 = 1 x 594, 2 x 297, 3 x 198, 6 x 99, 9 x 66, 11 x 54, 18 x 33, or 22 x 27
  • Taking the factor pair with the largest square number factor, we get √594 = (√9)(√66) = 3√66 ≈ 24.372115

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594 Factors

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