A Multiplication Based Logic Puzzle

Posts tagged ‘prime’

What Kind of Prime Is 859?

A prime number is a positive number that has exactly two factors, one and itself. (One has only one factor, so it is not a prime number.)

  • 859 is the 149th prime number.

A twin prime is a set of two prime numbers in which the second prime number is two more that the first prime number.

  • 859 is the second prime number in the 34th twin prime: (857, 859).

A prime triplet is a set of three consecutive prime numbers in which the last number is six more than the first number. Prime triplets always contain a set of twin primes.

  • 859 is in the 27th and 28th prime triplets: (853, 857, 859) and (857, 859, 863).

A prime quadruplet is a set of four consecutive prime numbers in which the last number is eight more than the first number. Prime quadruplets always contain TWO sets of overlapping prime triplets.

  • Even though prime numbers (853, 857, 859, 863) contain two sets of overlapping prime triplets, they do NOT form a prime quadruplet because the last number is ten more than the first number. Other than (5, 7, 11, 13), all prime quadruplets are prime decades whose last digits are 1, 3, 7, and 9, in THAT order.

There are other prime constellations like prime quintuplets and prime sextuplets, but each of those has to contain a prime quadruplet in it, so 859 isn’t in any of those.

859÷4 = 214 R3. Since that wasn’t R1, we know that 859 is NOT the hypotenuse of ANY Pythagorean triples.

Now you know what kind of prime 859 is.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-853-863

  • 859 is a prime number.
  • Prime factorization: 859 is prime.
  • The exponent of prime number 859 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 859 has exactly 2 factors.
  • Factors of 859: 1, 859
  • Factor pairs: 859 = 1 × 859
  • 859 has no square factors that allow its square root to be simplified. √859 ≈ 29.3087

How do we know that 859 is a prime number? If 859 were not a prime number, then it would be divisible by at least one prime number less than or equal to √859 ≈ 29.3. Since 859 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 859 is a prime number.

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857 Why do Primitive Pythagorean Triple Do That?

29² + 4² = 857, so 857 is the hypotenuse of a Pythagorean triple that just happens to be primitive:

  • 232-825-857 which was calculated from 2(29)(4), 29² – 4², 29² + 4².

Now, one of the legs of a primitive Pythagorean triple will be divisible by 4. In this case that number is 232. As I indicated when I wrote 821 Why Do Primitive Pythagorean Triples Do That? The sum of the number divisible by 4 in a primitive Pythagorean triple and the hypotenuse always equals a perfect square. So 232 + 857 = 1089 = 33².

This week I noticed something about the OTHER leg that might be even more amazing! The other leg in the triple, 825, is not divisible by 4, but get this: One half of the sum of a primitive’s hypotenuse and that leg that is NOT divisible by 4 also equals a square number: For example, (857 + 825)/2 = 1681/2 = 841 = 29².

But not only that, one half of the DIFFERENCE of a primitive’s hypotenuse and that leg that is NOT divisible by 4 equals a perfect square, too: (857 – 825)/2 = 32/2 = 16 = 4².

Now 29² and 4² look a little familiar to me today. Hmm… We’ve seen those two numbers together earlier in this post!

It seems that for Primitive Pythagorean Triples, what goes around comes around!

Check out the sums and difference of these other primitive Pythagorean triples:

Go ahead, try it with any other primitive triple on this list or otherwise!

If we let s = √(Sum/2) and let d = √(Difference/2), then we can rewrite some Pythagorean triple like this:

  • s² – d², 2sd, s² + d², so s² and d² are REALLY significant perfect squares!

Besides being the hypotenuse of a primitive Pythagorean triple, here are some other facts about the number 857:

857 is the sum of three consecutive prime numbers: 281 + 283 + 293.

857 is also the sum of the nineteen prime numbers from 11 to 83.

  • 857 is a prime number.
  • Prime factorization: 857 is prime.
  • The exponent of prime number 857 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 857 has exactly 2 factors.
  • Factors of 857: 1, 857
  • Factor pairs: 857 = 1 × 857
  • 857 has no square factors that allow its square root to be simplified. √857 ≈ 29.27456

How do we know that 857 is a prime number? If 857 were not a prime number, then it would be divisible by at least one prime number less than or equal to √857 ≈ 29.3. Since 857 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 857 is a prime number.

 

Here’s another way we know that 857 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 4² = 857 with 29 and 4 having no common prime factors, 857 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √857 ≈ 29.3. Since 853 is not divisible by 5, 13, 17, or 29, we know that 857 is a prime number.

853 You Can Do This Puzzle!

This is a level 1 puzzle that is easier than even most other level 1 puzzles. You can do this puzzle! If mathematics makes you uncomfortable, you can still do this puzzle! Even if math class is your worse nightmare, you can complete this puzzle, and gain a little confidence. Go ahead, give it a try! Figure out where each number from one to ten goes in the top row and also in the first column so that the puzzle turns into a mixed-up multiplication table. It’s easier and far less time consuming than Sudoku. You CAN do this puzzle! Then, after you find all the factors, and are feeling really good about yourself, IF you want, you can fill in all the other cells of this mixed up multiplication table.

Print the puzzles or type the solution on this excel file: 10-factors-853-863

853 is a prime number that leaves a remainder of 1 when divided by 4, so 853 is the hypotenuse of a Pythagorean triple: 205-828-853.

23² + 18² = 853 so 205-828-853 can be calculated from 23² – 18², 2(23)(18), 23² + 18².

  • 853 is a prime number.
  • Prime factorization: 853 is prime.
  • The exponent of prime number 853 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 853 has exactly 2 factors.
  • Factors of 853: 1, 853
  • Factor pairs: 853 = 1 × 853
  • 853 has no square factors that allow its square root to be simplified. √853 ≈ 29.20616

How do we know that 853 is a prime number? If 853 were not a prime number, then it would be divisible by at least one prime number less than or equal to √853 ≈ 29.2. Since 853 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 853 is a prime number.

Here’s another way we know that 853 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 23² + 18² = 853 with 23 and 18 having no common prime factors, 853 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √853 ≈ 29.2. Since 853 is not divisible by 5, 13, 17, or 29, we know that 853 is a prime number.

 

821 Why Do Primitive Pythagorean Triples Do That?

The most famous Pythagorean triple is 3-4-5.

Perhaps you know that of the three numbers in EVERY Pythagorean triple at least one of them will be divisible by 3, at least one of them will be divisible by 4, and at least one of them will be divisible by 5. That’s obvious for triple 3-4-5.

Here’s another example: Pythagorean triple 11-60-61. Two of those numbers are prime numbers, yet 60 is divisible by 3, 4, AND 5.

If the Pythagorean triple is a primitive, something else amazing happens:

That’s amazing all by itself, but that’s only part of the picture. Let’s look at the complete picture specifically using the triple with hypotenuse 821:

  • 25² + 14² = 821
  • 429² + 700² = 821²
  • 429700821 can be calculated from 25² – 14², 2(25)(14), 25² + 14²
  • 821 + 700 = 1521
  • √1521 = 39 which is 25 + 14
  • 429 ÷ 39 = 11

WHY does this happen to primitive Pythagorean triples?

It happened for 429700821 because 700 82125² + 14² + 2(25)(14) = (25 + 14)².

After all (a + b)² = a² +2ab + b² is always true.

And it can be rearranged: (a + b)² = a² + b² +2ab.

You can use a similar proof whenever the element of the Primitive triple that is divisible by 4 can be expressed as 2ab.

************

But it is a different, and perhaps simpler, story for many triples such as 94041 which was calculated using (9)(1), (9² – 1²)/2, (9² + 1²)/2.

In that case 40 + 41(9² – 1²)/2 + (9² + 1²)/2 = (9² – 1² + 9² + 1²)/2 = 9².

You can use a similar proof whenever the element of the Primitive triple that is divisible by 4 can be expressed as (a² – b²)/2.

************

Go ahead, take ANY primitive Pythagorean triple. Add the leg that is divisible by 4 to its hypotenuse. You will get a perfect square. Here’s a few more examples:

Similarly primitive triples that have been multiplied by a square number will also produce a perfect square, but you’ll have to be careful which leg you add to the hypotenuse if you multiplied by a square number that is a multiple of 4.

For example, 9(3-4-5) = 27-36-45 and 36 + 45 = 81, a square number.

But 4(3-4-5) = 12-16-20 so each number is divisible by 4. Note that 16 + 20 = 36, a square number, but 12 + 20 does not.

Here’s a few other essential facts about the number 821:

All of the odd numbers between 820 and 830, except 825, are prime numbers. Thus 821, 823, 827, 829 is the fourth prime decade.

  • 821 is a prime number.
  • Prime factorization: 821 is prime.
  • The exponent of prime number 821 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 821 has exactly 2 factors.
  • Factors of 821: 1, 821
  • Factor pairs: 821 = 1 x 821
  • 821 has no square factors that allow its square root to be simplified. √821 ≈ 28.65309756.

How do we know that 821 is a prime number? If 821 were not a prime number, then it would be divisible by at least one prime number less than or equal to √821 ≈ 28.7. Since 821 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 821 is a prime number.

Here’s another way we know that 821 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 14² = 821 with 25 and 14 having no common prime factors, 821 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √821 ≈ 28.7. Since 821 is not divisible by 5, 13, or 17, we know that 821 is a prime number.

 

 

797 and Mathematical Ways to Love

Mathwithbaddrawings.com has some thoughtful and entertaining Ways to Tell a Mathematician that you love them.

Artful Maths wrote a post that includes beautiful mathematical origami valentines and a “string art” cardioid that is made with a pencil instead of string.

This puzzle could be another mathematical way to show some love:

797-love-puzzle

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

Here are a few facts about the number 797:

797 is a palindrome in three bases:

  • 797 BASE 10 because 7(100) + 9(10) + 7(1) = 797
  • 565 BASE 12 because 5(144) + 6(12) + 5(1) = 797
  • 494 BASE 13 because 4(169) + 9(13) + 4(1) = 797

But there’s one more palindromic fact about the number 797: It is the sum of two square numbers that are also palindromes!

  • 797 = 121 + 676. Note that 11² = 121 and 26² = 676.

Since it is the sum of two squares, 797 will also be the hypotenuse a Pythagorean triple:

  • 555 – 572 – 797 calculated from 26² – 11², 2(26)(11), 26² + 11².

797 is the sum of three squares seven different ways:

  • 28² + 3² + 2² = 797
  • 27² + 8² + 2² = 797
  • 24² + 14² + 5² = 797
  • 24² + 11² + 10² = 797
  • 22² + 13² + 12² = 797
  • 21² + 16² + 10² = 797
  • 20² + 19² + 6² = 797

797 is also the sum of the 15 prime numbers from 23 to 83:

  • 23 + 29 + 31 + 37+ 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 797

Here is the most basic information about the number 797:

  • 797 is a prime number.
  • Prime factorization: 797 is prime.
  • The exponent of prime number 797 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 797 has exactly 2 factors.
  • Factors of 797: 1, 797
  • Factor pairs: 797 = 1 x 797
  • 797 has no square factors that allow its square root to be simplified. √797 ≈ 28.231188.

How do we know that 797 is a prime number? If 797 were not a prime number, then it would be divisible by at least one prime number less than or equal to √797 ≈ 28.2. Since 797 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 797 is a prime number.

Here’s another way we know that 797 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 11² = 797 with 26 and 11 having no common prime factors, 797 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √797 ≈ 28.2. Since 797 is not divisible by 5, 13, or 17, we know that 797 is a prime number.

797-factor-pairs

 

761 and Level 1

  • 761 is a prime number.
  • Prime factorization: 761 is prime.
  • The exponent of prime number 761 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 761 has exactly 2 factors.
  • Factors of 761: 1, 761
  • Factor pairs: 761 = 1 x 761
  • 761 has no square factors that allow its square root to be simplified. √761 ≈ 27.586228.

761-factor-pairs

How do we know that 761 is a prime number? If 761 were not a prime number, then it would be divisible by at least one prime number less than or equal to √761 ≈ 27.6. Since 761 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 761 is a prime number.

There will be a lot about squares in today’s post, and all of the clues in today’s puzzle are perfect squares, too:

761 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s more about the number 761:

Because 761 is a prime number, there is only one way to write it as the sum of consecutive numbers: 380 + 381 = 761.

However, because it is a centered square number, 761 can be written as the sum of consecutive square numbers: 20² + 19² = 761.

761 Centered Square Number

And because it is the sum of two consecutive square numbers, it is also the hypotenuse of a primitive Pythagorean triple: 39-760-761 which was calculated using 20² – 19², 2(20)(19), 20² + 19².

Notice also that 20 + 19 = 39, ⌊39²/2⌋ = 760, and ⌈39²/2⌉ = 761. (Those peculiar parenthesis mean⌊round down to the closest whole number⌋ and ⌈round up to the closest whole number⌉.)

In the 39 x 39 graphic shown above there are 760 white squares and 761 squares with some color. Thus, looking at that graphic is one way to look at the primitive Pythagorean triple 39-760-761.

All of that gives us another way to know that 761 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 20² + 19² = 761 with 20 and 19 having no common prime factors, 761 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √761 ≈ 27.6. Since 761 is not divisible by 5, 13, or 17, we know that 761 is a prime number.

761 can also be expressed as the sum of three squares TEN ways:

  • 27² + 4² + 4² = 761
  • 26² + 9² + 2² = 761
  • 26² + 7² + 6² = 761
  • 25² + 10² + 6² = 761
  • 24² + 13² + 4² = 761
  • 24² + 11² + 8² = 761
  • 23² + 14² + 6² = 761
  • 22² + 14² + 9² = 761
  • 21² + 16² + 8² = 761
  • 19² + 16² + 12² = 761

761 is a palindrome in two different bases:

  • 535 BASE 12; note that 5(144) + 3(12) + 5(1) = 761.
  • 1I1 BASE 20 (I = 18 base 10); note that 1(400) + 18(20) + 1(1) = 761.

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761 Factors

757 and Level 4

  • 757 is a prime number.
  • Prime factorization: 757 is prime.
  • The exponent of prime number 757 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 757 has exactly 2 factors.
  • Factors of 757: 1, 757
  • Factor pairs: 757 = 1 x 757
  • 757 has no square factors that allow its square root to be simplified. √757 ≈ 27.51363.

757-factor-pairs

How do we know that 757 is a prime number? If 757 were not a prime number, then it would be divisible by at least one prime number less than or equal to √757 ≈ 27.5. Since 757 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 757 is a prime number.

Here’s today’s puzzle. A logical way to solve it can be found in a table at the end of the post.

757 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s another way we know that 757 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 9² = 757 with 26 and 9 having no common prime factors, 757 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √757 ≈ 27.5. Since 757 is not divisible by 5, 13, or 17, we know that 757 is a prime number.

757 is prime so the only way it can be written as the sum of consecutive numbers is 378 + 379 = 757.

757 is the sum of seven consecutive prime numbers: 97 + 101 + 103 + 107 + 109 + 113 + 127 = 757.

From Stetson.edu we learn this distinguishing fact about the number 757:

757 reciprocal

Here are some square facts about the number 757:

26² + 9² = 757, and 468² + 595² = 757². That primitive Pythagorean triple, 468-595-757, can be calculated from 2(26)(9), 26² – 9², 26² + 9².

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

  • 24² + 10² + 9² = 757
  • 18² + 17² + 12² = 757

Finally 757 is a palindrome in four different bases:

  • 1001001 BASE 3; note that 1(3^6) + 0(3^5) + 0(3^4) + 1(3^3) + 0(3^2) + 0(3^1) + 1(3^0) = 757.
  • 757 BASE 10; note that 7(100) + 5(10) + 7(1) = 757.
  • 1F1 BASE 21 (F is 15 base 10); note that 1(21²) + 15(21) + 1(1) = 757.
  • 111 BASE 27; note that 1(27²) + 1(27) + 1(1) = 757.

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757 Logic

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