A Multiplication Based Logic Puzzle

Posts tagged ‘Prime factorization’

800 Which Pony Will Take Second Place?

Every 100 posts I summarize the amount of factors of the previous 100 numbers.

MANY of the numbers from 701 to 800 have FOUR factors, and any other number-of-factors doesn’t even come close. For this Horse Race, SECOND place is much more interesting as there are several lead changes. I’ve shorten the track so the second place number-of-factors can reach the finish line.

So go ahead, pick the number-of-factors pony you think will come in SECOND place. Your best bets are 2, 6, 8, 12, 16 OR the second row of 4 factors!

Make your selection, then click on the graphic below to see how your pony does!

800-horse-race-01

Now let me tell you a little bit about the number 800.

800-prime-factorization

  • 800 is a composite number.
  • Prime factorization: 800 = 2 x 2 x 2 x 2 x 2 x 5 x 5, which can be written 800 = (2^5) x (5^2)
  • The exponents in the prime factorization are 5 and 2. Adding one to each and multiplying we get (5 + 1)(2 + 1) = 6 x 3 = 18. Therefore 800 has exactly 18 factors.
  • Factors of 800: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800
  • Factor pairs: 800 = 1 x 800, 2 x 400, 4 x 200, 5 x 160, 8 x 100, 10 x 80, 16 x 50, 20 x 40 or 25 x 32
  • Taking the factor pair with the largest square number factor, we get √800 = (√400)(√2) = 20√2 ≈ 28.28427.

800-factor-pairs

800 is the sum of four consecutive primes:

  • 193 + 197 + 199 + 211 = 800

800 is a palindrome in three different bases.

  • 2222 BASE 7 because 2(7^3) + 2(49) + 2(7) + 2(1) = 800
  • 242 BASE 19 because 2(19²) + 4(19) + 2(1) = 800
  • PP BASE 31 (P is 25 base 10) because 25(31) + 25(1) = 800

800 is the sum of two squares two different ways:

  • 28² + 4² = 800
  • 20² + 20² = 800

That being true, it follows that 800 is the hypotenuse of two Pythagorean triples:

  • 480-640-800 which is 160 times 3-4-5
  • 224-768-800 which is 32 times 7-24-25

 

800 is also the sum of three squares:

  • 20² + 16² + 12² = 800

This chart summarizes the number of factors for the first 800 numbers and indicates that 39% of those numbers have square roots that can be simplified (reduced).

800-totals

In case you didn’t click on the Horse Race image before, here it is, no clicking required:

800 Horse Race

make science GIFs like this at MakeaGif

 

Advertisements

799 A Rose for Your Valentine

Roses are beautiful and make lovely gifts for Valentines or any other occasion. A Native American legend explains why roses have thorns.

The rose on today’s puzzle has thorns because without thorny clue 60 the puzzle would not have a unique solution. You can be sure that 60 will play an important part in using logic to find the solution to this puzzle.

With or without a valentine, love your brain and give the puzzle a try. It won’t be easy, but you should eventually be able to figure it out. My blogging friend, justkinga, has some other suggestions to show YOURSELF some love on Valentine’s Day.

799-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-795-799

7 + 9 + 9 = 25, a composite number.

7^3 + 9^3 + 9^3 = 1801, a prime number.

Stetson.edu states that 799 is the smallest number whose digits add up to a composite number AND whose digits cubed add up to a prime number.

It may seem like an improbable number fact, but it wasn’t too difficult to verify, and it really is true!

799 is also the smallest number whose digits add up to 25. (The digits of 889 also add up to 25, and its digits cubed also add up to a prime number. Could this be more than a coincidence?)

Here’s more about the number 799:

799 is palindrome 1H1 in BASE 21 (H is 17 base 10). Note that 1(441) + 17(21) + 1(1) = 799.

799 is the hypotenuse of Pythagorean triple 376-705-799 which is 47 times 8-15-17.

  • 799 is a composite number.
  • Prime factorization: 799 = 17 x 47
  • 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 799 has exactly 4 factors.
  • Factors of 799: 1, 17, 47, 799
  • Factor pairs: 799 = 1 x 799 or 17 x 47
  • 799 has no square factors that allow its square root to be simplified. √799 ≈ 266588.

799-factor-pairs

 

 

 

798 Cupid’s Arrow and Target

Here are two puzzles that go together and yet look out of sync. Sometimes cupid’s arrow reaches its target, and sometimes it doesn’t.

798-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-795-799

Here’s a little about the number 798:

798 is made from three consecutive numbers (7, 8, and 9), so it is divisible by three. The middle number, 8, is not divisible by three, so 798 is NOT divisible by nine.

798 is a palindrome in two bases:

  • 666 BASE 11 because 6(121) + 6(11) + 6(1) = 798
  • 383 BASE 15 because 3(225) + 8(15) + 3(1) = 798

798 is also the sum of two consecutive prime numbers: 397 + 401 = 798.

798 can be written as the sum of three squares four different ways:

  • 26² + 11² + 1² = 798
  • 25² + 13² + 2² =798
  • 23² + 13² + 10² = 798
  • 22² + 17² + 5² = 798

Here is 798’s factoring information:

  • 798 is a composite number.
  • Prime factorization: 798 = 2 x 3 x 7 x 19
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 798 has exactly 16 factors.
  • Factors of 798: 1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798
  • Factor pairs: 798 = 1 x 798, 2 x 399, 3 x 266, 6 x 133, 7 x 114, 14 x 57, 19 x 42, or 21 x 38
  • 798 has no square factors that allow its square root to be simplified. √798 ≈ 28.24889.

798-factor-pairs

796 and a Valentine

Here’s a puzzle you can print, cut out, and give as a valentine:

796-puzzle

You may know the divisibility rules for these powers of two:

  • If the last digit of a number is divisible by 2, the whole number is divisible by 2.
  • If the last two digits are divisible by 4, the whole number is divisible by 4.
  • If the last three digits are divisible by 8, the whole number is divisible by 8.

But I’m going to apply some other time-saving but possibly more confusing divisibility rules to the number 796:

  • 796 is divisible by 2 because 6 is an even number.
  • 796 is divisible by 4 because even number 6 is NOT divisible by 4, and 9 is an odd number.
  • 796 is NOT divisible by 8 because 96 is divisible by 8, and 7 is an odd number.

Because 796 is divisible by 4 but not by 8, it can be written as the sum of 8 consecutive numbers:

  • 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103 = 796

796 is also the sum of all the prime numbers from 113 to 149:

  • 113 + 127 + 131 + 137 + 139 + 149 = 796

Here is the factoring information for 796:

  • 796 is a composite number.
  • Prime factorization: 796 = 2 x 2 x 199, which can be written 796 = (2^2) x 199
  • 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 796 has exactly 6 factors.
  • Factors of 796: 1, 2, 4, 199, 398, 796
  • Factor pairs: 796 = 1 x 796, 2 x 398, or 4 x 199
  • Taking the factor pair with the largest square number factor, we get √796 = (√4)(√199) = 2√199 ≈ 28.21347.

796-factor-pairs

 

795 a Level 1 Puzzle with ♥

I learned yesterday that pushing ALT on the computer key pad and 3 on the number pad at the same time produces a when the keys are released. I added a few ‘s to today’s puzzle because Valentine’s day is almost here:

795-puzzle

  • 795 is a composite number.
  • Prime factorization: 795 = 3 x 5 x 53
  • 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 795 has exactly 8 factors.
  • Factors of 795: 1, 3, 5, 15, 53, 159, 265, 795
  • Factor pairs: 795 = 1 x 795, 3 x 265, 5 x 159, or 15 x 53
  • 795 has no square factors that allow its square root to be simplified. √795 ≈ 28.195744.

795-factor-pairs

Stetson.edu informs us that if we add up all the factors of 795 we get a number that is a perfect 4th power:

  • 1 + 3 + 5 + 15 + 53 + 159 + 265 + 795 = 1296, which happens to be the 4th power of 6. (Of course 1296 also equals 36 squared.)

795 is the hypotenuse of four Pythagorean triples and thus:

  • 69 – 792 – 795 which is 3 times (23 – 264 – 265)
  • 288 – 741 – 795 which is 3 times (96 – 247 – 265)
  • 420 – 675 – 795 which is 15 times (28 – 45 – 53)
  • 477 – 636 – 795 which is 159 times (3 – 4 – 5)

795 is also the sum of three squares two different ways:

  • 25² + 13² + 1² = 795
  • 25² + 11² + 7² = 795

 

 

791 and Level 4

To see if 791 is divisible by 7, you could try either one of these divisibility tricks:

  • 791 is divisible by 7 because 79 – 2(1) = 77 which obviously is divisible by 7.
  • 791 is divisible by 7 because 79 + 5(1) = 84 which most people know is 12 × 7.

791-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

————————————–

  • 791 is a composite number.
  • Prime factorization: 791 = 7 x 113
  • 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 791 has exactly 4 factors.
  • Factors of 791: 1, 7, 113, 791
  • Factor pairs: 791 = 1 x 791 or 7 x 113
  • 791 has no square factors that allow its square root to be simplified. √791 ≈ 28.12472222.

791-factor-pairs

790 and Level 3

790’s prime factorization is 2 × 5 × 79. Since two of those numbers, 5 and 79, have a remainder of one when divided by four, I automatically know that 790 is the hypotenuse of exactly four Pythagorean triples.

790-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

————————————–

  • 790 is a composite number.
  • Prime factorization: 790 = 2 x 5 x 79
  • 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 790 has exactly 8 factors.
  • Factors of 790: 1, 2, 5, 10, 79, 158, 395, 790
  • Factor pairs: 790 = 1 x 790, 2 x 395, 5 x 158, or 10 x 79
  • 790 has no square factors that allow its square root to be simplified. √790 ≈ 28.106939.

790-factor-pairs

Tag Cloud