prime

1789 An Easy Way to Solve This Subtraction Distraction Puzzle

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Today’s Puzzle:

To begin, I want to find a set of twelve consecutive numbers that will make the puzzle work. I want all of those numbers to be positive and relatively small. Thus, I would want to put a 12 in one of the last two boxes. Since the last triangle is -9, I would put the 12 in the last box. (If it were +9, I would put the 12 in the next to the last box.) Then I would do the following calculations based on the numbers in the triangles from right to left:

  • 12
  • 12 – 9 = 3
  • 3 + 4 = 7
  • 7 – 3 = 4
  • 4 + 4 = 8
  • 8 – 2 = 6
  • 6 – 1 = 5
  • 5 + 8 = 13
  • 13 – 4 = 9

And I would put the answers in the boxes from right to left:

The empty triangle makes me have to stop. Now I know I have to make some adjustments because one of the boxes has a 13 in it, but how much do I need to adjust each of those numbers? To answer that question, I will note what numbers from 1 to 12 are missing. I am missing 1, 2, 10, 11. The 13 I have means I can’t have the 1. I next access which of those missing numbers will yield -1. I note that 10 – 11 = -1, and write those numbers above the -1 triangle.

That leaves only the number 2 to place, but 10 + 4 ≠ 2, but 14. I place the 14 instead of the 2.

Now I have the twelve consecutive numbers from 3 to 14 in the boxes. If I subtract 2 from each of those twelve numbers, I will have all the numbers from 1 to 12. Also, it is easy to see that the number missing from the empty triangle is 2 whether I use 11 – 9 or 9 – 7.

Factors of 1789:

  • 1789 is a prime number.
  • Prime factorization: 1789 is prime.
  • 1789 has no exponents greater than 1 in its prime factorization, so √1789 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1789 has exactly 2 factors.
  • The factors of 1789 are outlined with their factor pair partners in the graphic below.

How do we know that 1789 is a prime number? If 1789 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1789. Since 1789 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1789 is a prime number.

More About the Number 1789:

1789 is the sum of two squares:
42² + 5² = 1789.

1789 is the hypotenuse of a Pythagorean triple:
420-1739-1789 calculated from 2(42)(5), 42² – 5², 42² + 5².

Here’s another way we know that 1789 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 42² + 5² = 1789 with 42 and 5 having no common prime factors, 1789 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1789. Since 1789 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1789 is a prime number.

1789 is in the twin prime 1787, 1789.

1789 is in the prime triplet 1783, 1787, 1789.

The first four multiples of 1789 are 1789, 3578, 5367, and 7156. Each of those multiples contains a 7. OEIS.org informs us that 1789 is the smallest number that can make that claim.

1789 is 414 in base 21 because 4(21²) + 1(21) + 4(1) = 1789.

 

1787 The11-Digit Palindromes of Base 2

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Today’s Puzzle:

1787 is an 11-digit palindrome in base 2. I wondered how many 11-digit palindromes there are, what they are, and what numbers they represent in base ten. I decided to try to make you wonder about all that as well. Try it out yourself before you read how I solved this puzzle.

The only digits in base 2, are 0 and 1. The first digit of any number must be 1 or else the number will not have eleven digits. The last digit also must be one for the number to be a palindrome. In fact, all five last digits will be determined by the first five digits. Thus, we only need to find all possible combinations of 0 and 1 that can occur in the second through sixth positions. There are 2⁵ ways to write 0 and 1 in those 5 positions. That means we know right away that there are 32 different 11-digit palindromes in base 2. I opened Excel and wrote those 32 different 11-digit numbers beginning with 00000 and ending with 11111. I put a 1 in front of them and had Excel copy the appropriate numbers into the last 5 spots. That gave me all the 11-digit palindromes. Then I had Excel multiply the values in each cell with the powers of 2 that head up each column to give the base 10 representations. This chart was the final product.

Did you notice that the first base 10 number in the chart is the number just after 2¹º and the last number is the number right before 2¹¹?

Factors of 1787:

  • 1787 is a prime number.
  • Prime factorization: 1787 is prime.
  • 1787 has no exponents greater than 1 in its prime factorization, so √1787 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1787 has exactly 2 factors.
  • The factors of 1787 are outlined with their factor pair partners in the graphic below.

How do we know that 1787 is a prime number? If 1787 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1787. Since 1787 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1787 is a prime number.

More About the Number 1787:

1787 and 1789 are twin primes.
1783, 1787, and 1789 are a prime triplet.

1787 is a palindrome in some other bases, too!
It’s 919 in base 14 because 9(14²) + 1(14) + 9(1) = 1787,
595 in base 18 because 5(18²) + 9(18) + 5(1) = 1787, and
191 in base 38 because 1(38²) + 9(38) + 1(1) = 1787.

1783 Another Mystery Puzzle

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Today’s Puzzle:

Can you find the factors for this mystery-level puzzle? There is only one solution.

Factors of 1783:

  • 1783 is a prime number.
  • Prime factorization: 1783 is prime.
  • 1783 has no exponents greater than 1 in its prime factorization, so √1783 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1783 has exactly 2 factors.
  • The factors of 1783 are outlined with their factor pair partners in the graphic below.

How do we know that 1783 is a prime number? If 1783 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1783. Since 1783 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1783 is a prime number.

More About the Number 1783:

1783 is palindrome 1L1 in base 33 because
1(33²) + 21(33) + 1(1) = 1783.

1783 is the sum of two consecutive numbers:
891 + 892 = 1783.

1783 is the difference of the squares of those same two consecutive numbers:
892² – 891² = 1783.
Of course, every other odd number can make a similar claim.

1777 A Different Heart

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Today’s Puzzle:

Every year I make some heart-shaped puzzles, but this heart is different: I haven’t used this design before. Can it win you over? Some of the clues are tricky, so make sure you use logic to find the one and only solution.

Factors of 1777:

  • 1777 is a prime number.
  • Prime factorization: 1777 is prime.
  • 1777 has no exponents greater than 1 in its prime factorization, so √1777 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1777 has exactly 2 factors.
  • The factors of 1777 are outlined with their factor pair partners in the graphic below.

How do we know that 1777 is a prime number? If 1777 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1777. Since 1777 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1777 is a prime number.

More About the Number 1777:

1777 is the sum of two squares:
39² + 16² = 1777.

1777 is the hypotenuse of a primitive Pythagorean triple:
1248-1265-1777, calculated from 2(39)(16), 39² – 16², 39² + 16²

Here’s another way we know that 1777 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 39² + 16² = 1777 with 39 and 16 having no common prime factors, 1777 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1777. Since 1777 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1777 is a prime number.

1777 looks interesting in some other bases:
It’s 12121 in base 6 because 1(6⁴) + 2(6³) + 1(6²) + 2(6¹) + 1(6º) = 1777,
2L2 in base 25, because 2(25²) + 21(25) + 2(1) = 1777, and
1B1in base 37, because 1(37²) + 11(37) + 1(1) = 1777.

1747 Getting a Super Bowl Ring Isn’t Easy

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Today’s Puzzle:

On Wednesday I read an article in my local paper titled From undrafted free agent to the biggest stage: Britain Covey’s ‘unique ride’ to Super Bowl LVII. It got me quite excited for today’s game. Unfortunately, between Wednesday and Sunday, Covey suffered a hamstring injury.  Getting a Super Bowl Ring certainly isn’t easy. Being able to contribute to your team’s winning the game isn’t easy either. UPDATE: Good News! He was able to make at least one play in the second quarter! Further update: Even though his team lost the game in the final minutes, this rookie played well when he was on the field. I think both teams played exceptionally well and gave us all an exciting game to watch.

Before the game, it is a mystery which team will win the game. The difficulty level of this puzzle is a mystery also.

If you look at today’s puzzle just right, I think it looks a little like a super bowl ring, but I forwarn you, it will not be easy to get this puzzle either. You will need to place the numbers from 1 to 12 both in the first column and in the top row so that the given clues are the products of the numbers you place. Logic and practice will get you there. Good Luck!

You also might enjoy this next Super Bowl puzzle I saw on Twitter:

Factors of 1747:

  • 1747 is a prime number.
  • Prime factorization: 1747 is prime.
  • 1747 has no exponents greater than 1 in its prime factorization, so √1747 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1747 has exactly 2 factors.
  • The factors of 1747 are outlined with their factor pair partners in the graphic below.

How do we know that 1747 is a prime number? If 1747 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1747. Since 1747 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1747 is a prime number.

More About the Number 1747:

1747 is a palindrome in base 15.
It’s 7B7₁₅ because 7(15²) + 11(15) + 7(1) = 1747.

1741 What Do You Notice About the Remainders?

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Today’s Puzzle:

The table below is a list of the first 280 prime numbers. Below each prime number is its remainder when the prime number is divided by 6. What do you notice about the remainders? What do you wonder?

Did you notice a fun fact starting with the number 1741? I would like to thank OEIS.org for making me aware of it.

Factors of 1741:

  • 1741 is a prime number.
  • Prime factorization: 1741 is prime.
  • 1741 has no exponents greater than 1 in its prime factorization, so √1741 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1741 has exactly 2 factors.
  • The factors of 1741 are outlined with their factor pair partners in the graphic below.

How do we know that 1741 is a prime number? If 1741 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1741. Since 1741 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1741 is a prime number.

More About the Number 1741:

1741 is the sum of two squares:
30² + 29² = 1741.

Here’s another way we know that 1741 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 29² = 1741 with 30 and 29 having no common prime factors, 1741 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1741. Since 1741 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1741 is a prime number.

Since 29 and 30 are consecutive numbers, we have another fun fact:

And 1741 is the hypotenuse of a Pythagorean triple:
59-1740-1741, calculated from 30² – 29², 2(30)(29), 30² + 29².

1741 is also the difference of two consecutive squares:
871² – 870² = 1741.

1723 A Little Virgács

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Today’s Puzzle:

I haven’t blogged much this year so I guess I deserved a little bit of virgács in my shoes this morning. Mikulás (St. Nick) leaves virgács in the boots of naughty little boys or girls in the wee hours of December 6. Treats are for the good kids. What is virgács? It is small golden spray-painted twigs bound with some pretty red ribbon.  Of course, all children are sometimes naughty and sometimes nice, so they could all expect to get virgács along with their treats in their boots this morning.

You can solve this virgács puzzle by starting with the clues at the top of the grid, finding their factors, and working down the puzzle row by row until you have found all the factors. Each number from 1 to 10 must appear exactly one time in both the first column and the top row.

Factors of 1723:

  • 1723 is a prime number.
  • Prime factorization: 1723 is prime.
  • 1723 has no exponents greater than 1 in its prime factorization, so √1723 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1723 has exactly 2 factors.
  • The factors of 1723 are outlined with their factor pair partners in the graphic below.

How do we know that 1723 is a prime number? If 1723 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1723. Since 1723 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1723 is a prime number.

More About the Number 1723:

1723 and 1721 are twin primes.

1723 is the sum of consecutive numbers:
861 + 862 = 1723.

1723 is the difference of consecutive squares:
862² – 861² = 1723.

1721 A Gift With Multiple Treasures

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Today’s Puzzle:

‘Tis the season of giving, and here’s a gift with multiple treasures inside. Write the numbers from 1 to 10 in both the first column and the top row so that the given clues are the products of the numbers you write. After you find the factors, you can complete the puzzle by finding all of the products.

Factors of 1721:

  • 1721 is a prime number.
  • Prime factorization: 1721 is prime.
  • 1721 has no exponents greater than 1 in its prime factorization, so √1721 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1721 has exactly 2 factors.
  • The factors of 1721 are outlined with their factor pair partners in the graphic below.

How do we know that 1721 is a prime number? If 1721 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1721. Since 1721 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1721 is a prime number.

More About the Number 1721:

1721 is the sum of two squares:
40² + 11² = 1721.

Here’s another way we know that 1721 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 11² = 1721 with 40 and 11 having no common prime factors, 1721 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1721. Since 1721 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1721 is a prime number.

1721 and 1723 are twin primes.

1709 Sometimes “Guess and Check” Is a Good Strategy

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Today’s Puzzle:

Most of the puzzles I publish are logic puzzles, and I encourage you to find the logic of the puzzle and not guess and check.  However, guess and check is a legitimate strategy in mathematics, and it is a legitimate strategy to solve this particular puzzle.

Since one of the clues is -9, we know that the two boxes under it must be [1, 10], [2, 11], or [3, 12].

Suppose you assume it’s 1 – 10 = -9. If you fill out the rest of the boxes you would get:

You know that isn’t right because zero is not a number from 1 to 12. No problem. Simply add one to each of the numbers you wrote in, and the puzzle will be solved with only numbers from 1 to 12.

Suppose you assumed it’s 3 -12 = -9. The rest of the boxes would look like this:

Again, 13 is not included in the numbers from 1 to 12, but you can fix it by subtracting 1 from each of the numbers you wrote in. Easy Peasy.

Factors of 1709:

  • 1709 is a prime number.
  • Prime factorization: 1709 is prime.
  • 1709 has no exponents greater than 1 in its prime factorization, so √1709 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1709 has exactly 2 factors.
  • The factors of 1709 are outlined with their factor pair partners in the graphic below.

How do we know that 1709 is a prime number? If 1709 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1709. Since 1709 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1709 is a prime number.

More About the Number 1709:

1709 is the sum of two squares:
35² + 22² = 1709.

1709 is the hypotenuse of a Pythagorean triple:
741-1540-1709, calculated from 35² – 22², 2(35)(22), 35² + 22².

Here’s another way we know that 1709 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 22² = 1709 with 35 and 22 having no common prime factors, 1709 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1709. Since 1709 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1709 is a prime number.

1699 Sweet Candy Cane

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Today’s Puzzle:

Solving this candy cane puzzle can be a sweet experience. Just use logic to write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Factors of 1699:

  • 1699 is a prime number.
  • Prime factorization: 1699 is prime.
  • 1699 has no exponents greater than 1 in its prime factorization, so √1699 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1699 has exactly 2 factors.
  • The factors of 1699 are outlined with their factor pair partners in the graphic below.

How do we know that 1699 is a prime number? If 1699 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1699. Since 1699 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1699 is a prime number.

More About the Number 1699:

1699 is the third prime in a prime triple. What are the other two primes in that triple?

1699 is the difference of two squares:
850² – 849² = 1699.