602 and Level 1

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(5^602) is a very large number with 421 digits. What are the first three of those 421 digits? 602. Thank you, OEIS.org for that interesting number fact.

602 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 602 is a composite number.
  • Prime factorization: 602 = 2 x 7 x 43
  • 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 602 has exactly 8 factors.
  • Factors of 602: 1, 2, 7, 14, 43, 86, 301, 602
  • Factor pairs: 602 = 1 x 602, 2 x 301, 7 x 86, or 14 x 43
  • 602 has no square factors that allow its square root to be simplified. √602 ≈ 24.535688.

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602 Factors

 

601 The Other Side of a Twin Prime

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601 and 599 are twin primes because there is only one number between them. That number happens to be 600, a number with 24 factors.

Twin primes certainly are not identical twins. That solitary number between them and the fact that they are both primes is usually all they have in common.

Twin Onesies Raglan - Yes We Are Twins No We Are Not Identical - Funny Twin Onesies - Raglan Boy/Girl Twin Shirts - Twin Gifts Boy/Girl -

599 and 601 could wear these cute onesies. Click on the picture for the etsy link.

Other famous twin primes are 11 and 13, 17 and 19, 29 and 31, and 41 and 43.

3, 5, 7 is the only prime triplet that consists of three consecutive odd numbers because 3 is the only prime number divisible by 3, and all sets of three consecutive odd numbers must have at least one number in the set that is divisible by 3. Other than 2, 3, 5, all other so-called prime triplets come from four consecutive odd numbers. Three of the four numbers will be prime while either the second or the third number will not be.

Twin primes like 599 and 601 can be prime factors. That is something 600 can never do.

For example, 599 x 601 = 359, 999 which is 600² – 1. They are its prime factors. 600 is a factor of an infinite amount of numbers, but it is not a prime factor of any number.

What are some ways twin primes are different from each other? I’ll use twin primes 599 and 601 as examples.

When each prime in a twin prime is divided by 4, the remainder will be 1 for one of them and 3 for the other. The one that has a remainder of 1 will be the sum of two square numbers and will also be the hypotenuse of a primitive Pythagorean triple. The one with a remainder of 3 will never be either of those things. In the triple and twin primes listed above 5, 13, 17, 29, 41, and 601 have a remainder of 1 when divided by 4.

601 = 24² + 5², and is the hypotenuse of the primitive Pythagorean triple 480, 551, 601 while 599 doesn’t do either of those things.

599 is the smallest number whose digits add up to 23. If you add up 601’s digits, you’ll get 7. Figure out how many numbers less than 601 have digits that add up to 7. It’s a lot.

601 has accomplished some other impressive things:

601 Centered Pentagonal Number

1 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55 + 60 + 65 + 70 + 75 = 601 which makes 601 the 16th centered pentagonal number

Because it is a centered pentagonal number, 601 is one more than five times the 15th triangular number.

The factors of numbers from 601 to 607 form a symmetrical shape.

601 to 607 factors

I gave the factoring information for 599 previously. Here is the information for 601:

  • 601 is a prime number.
  • Prime factorization: 601 is prime.
  • The exponent of prime number 601 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 601 has exactly 2 factors.
  • Factors of 601: 1, 601
  • Factor pairs: 601 = 1 x 601
  • 601 has no square factors that allow its square root to be simplified. √601 ≈ 24.5153

How do we know that 601 is a prime number? If 601 were not a prime number, then it would be divisible by at least one prime number less than or equal to √601 ≈ 24.5. Since 601 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 601 is a prime number.

How Many Factors Do the Numbers Up to 600 Have?

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Not Much of a Horse Race:

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 square roots that can be simplified. 40% of these numbers do, but the total numbers from 1 to 600 that have simplifiable square roots is a little lower than that. You can see that in red 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

Factors of 600:

  • 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.)

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

600 has twelve factor pairs. One of the factor pairs adds up to 50, and a different one subtracts to 50. If you can identify those factor pairs, then you can solve the second puzzle!

More about the Number 600:

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?

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

/ ivasallay / 4 Comments

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, OEIS.org 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

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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 OEIS.org: (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

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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?

OEIS.org 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

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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

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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

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

(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

593 and Level 1

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593 is the sum of the seven prime numbers from 71 to 101 as well as the nine prime numbers from 47 to 83.

When the number formed from the last two digits of prime number 593 is divided by 4, the remainder is 1 so 593 can be expressed at the sum of two squares AND is the hypotenuse of a primitive Pythagorean triple. Specifically:

23² + 8² = 593

593 is the hypotenuse of the primitive Pythagorean triple prime 368-465-593

593 Puzzle

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

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  • 593 is a prime number.
  • Prime factorization: 593 is prime.
  • The exponent of prime number 593 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 593 has exactly 2 factors.
  • Factors of 593: 1, 593
  • Factor pairs: 593 = 1 x 593
  • 593 has no square factors that allow its square root to be simplified. √593 ≈ 24.35159

How do we know that 593 is a prime number? If 593 were not a prime number, then it would be divisible by at least one prime number less than or equal to √593 ≈ 24.4. Since 593 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 593 is a prime number.

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593 Factors