A Multiplication Based Logic Puzzle

805 and Level 4

23 × 35 = 805 so we shouldn’t be surprised that 805 is palindrome NN in BASE 34. N is the same as 23 in base 10. Thus NN can be derived from 23(34) + 23(1) = 23(34 + 1) = 23 × 35 = 805. NN obviously is divisible by 11 like all 2 digit palindromes are.

Since 23 = 22 + 1, should we expect that 805 is a palindrome in BASE 22? No, and that is for the same reason that not all multiples of 11 are palindromes.

Finding the factors in today’s puzzle shouldn’t be very difficult, but the last few might be trickier than the rest:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

  • 805 is a composite number.
  • Prime factorization: 805 = 5 x 7 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 805 has exactly 8 factors.
  • Factors of 805: 1, 5, 7, 23, 35, 115, 161, 805
  • Factor pairs: 805 = 1 x 805, 5 x 161, 7 x 115, or 23 x 35
  • 795 has no square factors that allow its square root to be simplified. √805 ≈ 28.37252

805 is the hypotenuse of a Pythagorean triple:

  • 483-644-805, which is 3-4-5 times 161

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

  • 25² + 12² + 6² = 805
  • 24² + 15² + 2² = 805
  • 20² + 18² + 9² = 805
  • 18² + 16² + 15² = 805

 

 

 

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Factor Rainbows can be a wonderful way to display the factors of a number. Not only are all the factors listed in order from smallest to greatest, but the factor pairs are joined together with the same color band.

The number 804 has 12 factors so it makes a lovely rainbow with 6 different color bands.

Is there a pot of gold at the end of this factor rainbow? I’ll let you decide the answer to that question.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Finding golden nuggets of information about a number might be less difficult than finding pots of gold.

I always begin the painstaking mining process by looking at the factors of the number:

  • 804 is a composite number.
  • Prime factorization: 804 = 2 x 2 x 3 x 67, which can be written 804 = (2^2) x 3 x 67
  • 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 804 has exactly 12 factors.
  • Factors of 804: 1, 2, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804
  • Factor pairs: 804 = 1 x 804, 2 x 402, 3 x 268, 4 x 201, 6 x 134, or 12 x 67
  • Taking the factor pair with the largest square number factor, we get √804 = (√4)(√201) = 2√201 ≈ 28.3548937575

About half of everything there was already in the factor rainbow.

Finding nuggets of information about the number 804 has been a little difficult and disappointing:

  • None of 804’s prime factors can be written as 4N+1, so 804 is NOT the hypotenuse of any Pythagorean triples.
  • 804 is NOT a palindrome in base 36 or any base less than that.
  • 804 is NOT the sum of any consecutive prime numbers.

Even though I did not find any golden nuggets in those places, I kept looking and finally found a couple of gems about the number 804:

804 can be written as the sum of three squares four different ways, and all of those ways have some definition of double in them:

  • 28² + 4² + 2² = 804
  • 26² + 8² + 8² = 804
  • 22² + 16² + 8² = 804
  • 20² + 20² + 2² = 804

Stetson.edu also gives us a nugget about the number 804 that may be a bit too heavy for most people to handle: “804 is a value of n for which 2φ(n) = φ(n+1).” That basically means that there are exactly half as many numbers less than 804 that are NOT divisible by its prime factors (2, 3, or 67) as there are numbers less than 805 that are NOT divisible by its prime factors (5, 7, or 23).

I started looking for golden specs about 804 in places that I don’t usually look.

267 + 268 + 269 = 804 so 804 is the sum of 3 consecutive numbers.

As stated before 804 is never the hypotenuse of a Pythagorean triple. However to find all the times it is a leg in a triple will require a lot of labor especially since 804 has so many factors, including 4, and two of its factor pairs have factors where both factors are even.

  • 134, 6, is an even factor pair, so (134 + 6)/2 = 70, and (134-6)/2 = 64. Thus 804 = 134·6 = (70 + 64)(70 – 64) = 70² – 64² .
  • 402, 2, is another even factor pair, so (402 + 2)/2 = 202, and (402 – 2)/2 = 200. Thus 804 = 402·2 = (200 + 2)(200 – 2) = 202² – 200²
  • Likewise odd or even factor pairs that make the factors of 804 can also be used to find Pythagorean triples.

So to find all Pythagorean triples that contain the number 804, we will have to find all the times 804 satisfies one of these FOUR conditions:

  1. 804 = 2k(a)(b) so that 804 is in the triple 2k(a)(b), k(a² – b²), k(a + b²) OR the triple k(a² – b²), 2k(a)(b), k(a + b²).
  2. 804 = 2(a)(b) so that 804 is in the triple 2(a)(b), a² – b², a + b² OR the triple a² – b², 2(a)(b), a + b².
  3. 804 = a² – b² so that 804 is in the triple a² – b², 2(a)(b), a + b² OR the triple 2(a)(b), a² – b², a + b².
  4. 804 = k(a² – b²) so that 804 is in the triple k(a² – b²), 2k(a)(b), k(a + b²) OR the triple 2k(a)(b), k(a² – b²), k(a + b²).

Let the mining process begin! I’ll list the triples with the shortest legs first and color code each triple according to the condition I used. Since Saint Patrick’s Day is almost here, I’m publishing this list even though I may not have found all of the blue triples yet. I’ll update the list if I find more.

  • 335-804-871 which used 804 = 2·67(3)(2) to make a triple that is 5-12-13 times 67
  • 603-804-1005 which used 804 = 2·201(2)(1) to make a triple that is 3-4-5 times 201
  • 804-1072-1340 which used 804 = 268(2² – 1²) to make a triple that  is 3-4-5 times 268
  • 804-2345-2479 which used 804 = 2·67(6)(1) to make a triple that is 12-35-37 times 67
  • 804-4453-4525, which used 804 = 2(6)(67)
  • 804-8960-8996, which used 804 = 70² – 64² or 804 = 4(35² – 32²) to make a triple that is 201-2240-2249 times
  • 804-17947-17965, which used 804 = 2(134)(3)
  • 804-26928-26940 which used 804 = 12(34² – 33²) to make a triple that is 67-2244-2245 times 12
  • 804-40397-40405, which used 804 = 2(201)(2)
  • 804-53865-53871 which used 804 = 2·3(134)(1) to make a triple that is 268-17955-17957 times 3
  • 804-80800-80804 which used 804 = 202² – 200²  or 804 = 4(101² – 100²) to make a triple that is 201-20200-20201 times 4
  • 804-161603-161605, a primitive Pythagorean triple, that used 804 = 2(402)(1)

If you look for a pot of gold at the end of a rainbow, you’re bound to be disappointed. Science/How Stuff Works just had to crush dreams and dispel 10 Myths About Rainbows. Unfortunately a pot of gold being at the rainbow’s end is included on that list. Still I suppose we could still put every golden spec or nugget about 804 into a little pot and call it a pot of gold.

Or if you are as clever and quick as a leprechaun, perhaps you will consider finding Pythagorean triples to be like finding pots of gold.

8 – 0 + 3 = 11, so 803 is divisible by 11.

  • 803 is a composite number.
  • Prime factorization: 803 = 11 x 73
  • 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 803 has exactly 4 factors.
  • Factors of 803: 1, 11, 73, 803
  • Factor pairs: 803 = 1 x 803 or 11 x 73
  • 803 has no square factors that allow its square root to be simplified. √803 ≈ 3372546

You can solve today’s Level 3 puzzle by starting at the top of the first column, finding the factors of the clues and writing them in the appropriate cells. Then continue to go down that same column, cell by cell, finding factors and writing them down until you reach the bottom. Make sure that both the first column and the top row have each number from 1 to 10 written in them.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Here’s a few more facts about the number 803:

803 is the hypotenuse of a Pythagorean triple:

  • 528-605-803 which is 11 times another Pythagorean triple: 48-55-73

803 is the sum of three squares six different ways:

  • 27² + 7² + 5² = 803
  • 25² + 13² + 3² = 803
  • 23² + 15² + 7² = 803
  • 21² + 19² + 1² = 803
  • 19² + 19² + 9² = 803
  • 17² + 17² + 15² = 803

803 is the sum of consecutive prime numbers three different ways. Prime factor 11 is not in any of those ways, but prime factor 73 is in two of them.

  • 263 + 269 + 271 = 803, that’s 3 consecutive primes.
  • 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 = 803, that’s 9 consecutive primes.
  • 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 803, that’s 13 consecutive primes.

803 is a palindrome in two bases. Why are the numbers similar in these two palindromes?

  • 30203 BASE 4 because 3(256) + 0(64) + 2(16) + 0(4) + 3(1) = 803
  • 323 BASE 16 because 3(16²) + 2(16) + 3(1) = 803

 

 

 

In the United States tomorrow’s date is written 3-14. Because 3.14 is a famous approximation for π (pi), people all over the country will eat pie to celebrate Pi Day. This afternoon I took a picture of this sign and the pie display at my local Smith’s Food and Drug.

I took that picture right when I walked into the store, but there were no pies on display for National Pi Day.

About 15 minutes later I returned to the display to take another picture. Now there were pies on the table! I told a salesperson who I think worked on the display that I was going to take a picture and put it on my blog. She asked what kind of a blog I wrote. I told her a math blog. She looked puzzled and asked why I would want to put a picture of pies on a math blog. Then she turned around, looked at the display, and said something like, “Oh, now I get it, the number pi.”

How do you choose between apple, cherry, or peach pie? It’s much easier if you choose two and then you can get a free 8 oz. Cool Whip, too. Yummy.

If by chance you prefer pizza pi, here’s a thought from twitter that is often repeated in March:

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And here’s some original artwork that displays pi in a way I had never thought of before:

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BREAKING: secret of Pi revealed #PiDay pic.twitter.com/Ao8BQp31jd

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You can also look here for a million digits of pi.

But pi is not the only interesting number in the world. Every number has its own curiosities. Let me tell you some reasons to get excited about the number 802:

802 is the sum of two squares:

  • 21² + 19² = 802

So 802 is the hypotenuse of a Pythagorean triple:

  • 80-798-802, which is 2 times another triple: 40-399-401.

It also means something else: Since odd numbers 21 and 19 have no common prime factors, 802 can be evenly divide by 2. Duh. . ., but it also means that unless 802 is also divisible by 5, 13, or 17, its only factors will be 2 and a prime number! Why are those three numbers the only ones I care about? Because they are the only prime number Pythagorean triple hypotenuses less than √802 ≈ 28.3.

Guess what? 5, 13, and 17 do not divide evenly into 802, so 802 is the product of 2 and a prime number which happens to be 401.

  • 802 is a composite number.
  • Prime factorization: 802 = 2 x 401
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 802 has exactly 4 factors.
  • Factors of 802: 1, 2, 401, 802
  • Factor pairs: 802 = 1 x 802 or 2 x 401
  • 802 has no square factors that allow its square root to be simplified. √802 ≈ 28.3196045

Today’s puzzle is number 802 to distinguish it from every other puzzle I’ve made. Writing the numbers 1 – 10 in both the top row and the first column so that the factors and the clues work together as a multiplication table is as easy as pie!

Print the puzzles or type the solution on this excel file: 10-factors 801-806

And here is a little more about the number 802:

802 is the sum of 8 consecutive prime numbers:

  • 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 802

802 can also be written as the sum of three squares three different ways:

  • 28² + 3² + 3² = 802
  • 27² + 8² + 3² = 802
  • 24² + 15² + 1² = 802

802 is also a palindrome in two other bases:

  • 414 BASE 14 because 4(196) + 1(14) + 4(1) = 802
  • 202 BASE 20 because 2(400) + 0(20) + 2(1) = 802

 

When it comes to applying our tried and true trick for divisibility by nine to the number 801, zero is just a place holder. Thus, since 81 is divisible by 9, so is 801. Adding up its digits was hardly necessary.

  • 801 is a composite number.
  • Prime factorization: 801 = 3 x 3 x 89, which can be written 801 = (3^2) x 89
  • 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 801 has exactly 6 factors.
  • Factors of 801: 1, 3, 9, 89, 267, 801
  • Factor pairs: 801 = 1 x 801, 3 x 267, or 9 x 89
  • Taking the factor pair with the largest square number factor, we get √801 = (√9)(√89) = 3√89 ≈ 28.301943396.

Would you be surprised to know the following division facts?

  • 81 ÷ 3 = 27
  • 801 ÷ 3 = 267
  • 8001 ÷ 3 = 2667
  • 80001 ÷ 3 = 26667 and so forth. The number of 6’s in the quotient is the same as the number of 0’s in the dividend!

Here are some more predictable division facts:

  • 81 ÷ 9 = 9
  • 801 ÷ 9 = 89
  • 8001 ÷ 9 = 889
  • 80001 ÷ 9 = 8889 and so forth. You guessed it! The number of 8’s in the quotient is the same as the number of 0’s in the dividend!

Even though you can’t see 81 in this puzzle with all perfect square clues, it isn’t difficult to see where 9 × 9 and 81 belong:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

801 is a palindrome in three bases:

  • 1441 BASE 8 because 1(8^3) + 4(8^2) + 4(8) + 1(1) = 801
  • 2D2 BASE 17 D is 13 base 10 because 2(289) + 13(17) = 2(1) = 801
  • 171 BASE 25 because 1(25²) + 7(25) + 1(1) =801

801 is the sum of two squares:

  • 24² + 15² =801

So it follows that 801 is the hypotenuse of a Pythagorean triple:

  • 351-720-801 which is 9 times 39-80-89

801 is the sum of three squares TEN ways:

  1. 28² + 4² + 1² = 801
  2. 27² + 6² + 6² = 801
  3. 26² + 11² + 2² =801
  4. 26² + 10² + 5² = 801
  5. 24² + 12² + 9² = 801
  6. 23² + 16² + 4² = 801
  7. 22² + 14² + 11² = 801
  8. 21² + 18² + 6² = 801
  9. 20² + 20² + 1² = 801
  10. 17² + 16² + 16² = 801

Stetson.edu gives us this last fun fact:

801 = (7! + 8! + 9! + 10!) / (7 × 8 × 9 × 10).

 

 

 

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

 

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

 

 

 

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