1723 A Little Virgács

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

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

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

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.

1697 A Boot in the Window

Today’s Puzzle:

Tonight throughout many parts of the world children will place their polished boots in a window awaiting a visit from St. Nick. In the morning they will find their boots filled with favorite candies.

You can solve this boot puzzle by writing the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues will become the start of a multiplication table.

Factors of 1697:

  • 1697 is a prime number.
  • Prime factorization: 1697 is prime.
  • 1697 has no exponents greater than 1 in its prime factorization, so √1697 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1697 has exactly 2 factors.
  • The factors of 1697 are outlined with their factor pair partners in the graphic below.

How do we know that 1697 is a prime number? If 1697 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1697. Since 1697 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1697 is a prime number.

More About the Number 1697:

1697 is the sum of two squares:
41² + 4² = 1697.

1697 is the hypotenuse of a Pythagorean triple:
328-1665-1697, calculated from 2(41)(4), 41² – 4², 41² + 4².

Here’s another way we know that 1697 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 41² + 4² = 1697 with 41 and 4 having no common prime factors, 1697 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1697. Since 1697 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1697 is a prime number.

1693 Tricky Turkey

Today’s Puzzle:

You cannot gobble this turkey up unless you can find all of its factors!

Use logic and multiplication facts. It won’t be easy, but write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Here’s the same puzzle without the added color:

Factors of 1693:

  • 1693 is a prime number.
  • Prime factorization: 1693 is prime.
  • 1693 has no exponents greater than 1 in its prime factorization, so √1693 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1693 has exactly 2 factors.
  • The factors of 1693 are outlined with their factor pair partners in the graphic below.

How do we know that 1693 is a prime number? If 1693 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1693. Since 1693 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1693 is a prime number.

More About the Number 1693:

1693 is the sum of two squares:
37² + 18² = 1693.

1693 is the hypotenuse of a Pythagorean triple:
1045-1332-1693, calculated from 37² – 18², 2(37)(18), 37² + 18².

Here’s another way we know that 1693 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 37² + 18² = 1693 with 37 and 18 having no common prime factors, 1693 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1693. Since 1693 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1693 is a prime number.

1669 and Level 6

Today’s Puzzle:

Some sets of clues in this puzzle have two possible common factors, and another set has three possible common factors. Don’t guess which ones to use, but use logic instead! Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues form a multiplication table.

Factors of 1669:

  • 1669 is a prime number.
  • Prime factorization: 1669 is prime.
  • 1669 has no exponents greater than 1 in its prime factorization, so √1669 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1669 has exactly 2 factors.
  • The factors of 1669 are outlined with their factor pair partners in the graphic below.

How do we know that 1669 is a prime number? If 1669 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1669. Since 1669 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1669 is a prime number.

More About the Number 1669:

1669 is the sum of two squares:
38² + 15² = 1669.

1669 is the hypotenuse of a primitive Pythagorean triple:
1140-1219-1669, calculated from 2(38)(15), 38² – 15², 38² + 15².

Here’s another way we know that 1669 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 15² = 1669 with 38 and 15 having no common prime factors, 1669 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1669. Since 1669 is not divisible by 5, 13, 17, 29, or 37, we know that 1669 is a prime number.

1667 and Level 4

Today’s Puzzle:

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 function like a multiplication table.

Factors of 1667:

  • 1667 is a prime number.
  • Prime factorization: 1667 is prime.
  • 1667 has no exponents greater than 1 in its prime factorization, so √1667 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1667 has exactly 2 factors.
  • The factors of 1667 are outlined with their factor pair partners in the graphic below.

How do we know that 1667 is a prime number? If 1667 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1667. Since 1667 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1667 is a prime number.

More About the Number 1667:

Look at these consecutive number facts about the number 1667:
833 + 834 = 1667.
834² – 833² = 1667.

As the chart below shows, 1667 was ALMOST the fourth consecutive prime number ending in 7. Too bad prime number 1663 got in the way of that happening.

 

1663 and Level 1

Today’s Puzzle:

Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1663:

  • 1663 is a prime number.
  • Prime factorization: 1663 is prime.
  • 1663 has no exponents greater than 1 in its prime factorization, so √1663 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1663 has exactly 2 factors.
  • The factors of 1663 are outlined with their factor pair partners in the graphic below.

How do we know that 1663 is a prime number? If 1663 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1663. Since 1663 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1663 is a prime number.

More About the Number 1663:

1663 is the sum of consecutive numbers in only one way:
831 + 832 = 1663.

1663 is the difference of two squares in only one way:
832² – 831² = 1663.

What do you notice about those two number facts?

Prime Number 1657 is the 24th Centered Hexagonal Number

Today’s Puzzle:

Draw six triangles on the graphic below to show that 1657 is one more than 6 times the 23rd triangular number.

Factors of 1657:

  • 1657 is a prime number.
  • Prime factorization: 1657 is prime.
  • 1657 has no exponents greater than 1 in its prime factorization, so √1657 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1657 has exactly 2 factors.
  • The factors of 1657 are outlined with their factor pair partners in the graphic below.

How do we know that 1657 is a prime number? If 1657 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1657. Since 1657 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1657 is a prime number.

More About the Number 1657:

1657 is the sum of two squares:
36² + 19² = 1657.

1657 is the hypotenuse of a primitive Pythagorean triple:
935-1368-1657, calculated from 36² – 19², 2(36)(19), 36² + 19².

Here’s another way we know that 1657 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 36² + 19² = 1657 with 36 and 19 having no common prime factors, 1657 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1657. Since 1657 is not divisible by 5, 13, 17, 29, or 37, we know that 1657 is a prime number.

Do you notice anything else special about the number 1657 in this color-coded chart?