1621 Give This Purple Egg a Crack!

Today’s Puzzle:

Here’s a level-5 purple Easter egg for you to try. All you need to do is write the numbers from 1 to 12 in both the first column and in the top row so that those numbers and the given clues function like a multiplication table. Go ahead. Give it a crack!

Factors of 1621:

  • 1621 is a prime number.
  • Prime factorization: 1621 is prime.
  • 1621 has no exponents greater than 1 in its prime factorization, so √1621 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1621 has exactly 2 factors.
  • The factors of 1621 are outlined with their factor pair partners in the graphic below.

How do we know that 1621 is a prime number? If 1621 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1621. Since 1621 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1621 is a prime number.

More about the Number 1621:

OEIS.org informs us that 1621 is in an interesting group of prime numbers.

I have verified it. They really are all prime!

1621 is the sum of two squares:
39² + 10² = 1621.

1621 is the hypotenuse of a Pythagorean triple:
780-1421-1621, calculated from 2(39)(10), 39² – 10², 39² + 10².

Here’s another way we know that 1621 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 39² + 10² = 1621 with 39 and 10 having no common prime factors, 1621 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1621. Since 1621 is not divisible by 5, 13, 17, 29, or 37, we know that 1621 is a prime number.

1619 A Pink Egg Hidden in the Grass

Today’s Puzzle:

Easter is less than two weeks away. This pink puzzle is the first of three level-5 Easter eggs hidden amongst some blades of grass for you to find and solve. The puzzle might be a little tricky, but use logic every step of the way, and you’ll be able to find the unique solution:

Factors of 1619:

  • 1619 is a prime number.
  • Prime factorization: 1619 is prime.
  • 1619 has no exponents greater than 1 in its prime factorization, so √1619 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1619 has exactly 2 factors.
  • The factors of 1619 are outlined with their factor pair partners in the graphic below.

How do we know that 1619 is a prime number? If 1619 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1619. Since 1619 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1619 is a prime number.

More about the Number 1619:

1619 is the sum of two consecutive numbers:
809 + 810 = 1619.

1619 is also the difference of two consecutive squares:
810² – 809² = 1619.

What do you think about that?

1613 Comparing π and √10

Venn Diagram Pies:

In the United States many people celebrate pi day. This year it will be one hour shorter as we move to Daylight Saving Time. Since it will be on a Sunday, it might not get as much attention in school. Do we make too much of a deal about the number pi? It’s about 0.02 less than √10, an important, yet less-known number. I compare the two numbers in this Venn diagram:

We ought to take advantage of any reason to celebrate anything and everything in mathematics. I will be making some kind of pie to celebrate pi day, and I hope you do the same,

Now let’s move on to the ….

Factors of 1613:

  • 1613 is a prime number.
  • Prime factorization: 1613 is prime.
  • 1613 has no exponents greater than 1 in its prime factorization, so √1613 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1613 has exactly 2 factors.
  • The factors of 1613 are outlined with their factor pair partners in the graphic below.

How do we know that 1613 is a prime number? If 1613 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1613. Since 1613 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1613 is a prime number.

More about the Number 1613:

1613 is the sum of two squares:
38² + 13² = 1613.

1613 is the hypotenuse of a Pythagorean triple:
988-1275-1613, calculated from 2(38)(13), 38² – 13², 38² + 13².

Here’s another way we know that 1613 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 13² = 1613 with 38 and 13 having no common prime factors, 1613 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1613. Since 1613 is not divisible by 5, 13, 17, 29, or 37, we know that 1613 is a prime number.

 

1609 Pot of Gold

Today’s Puzzle:

They say at the end of the rainbow, there is a pot of gold that belongs to some leprechaun. Because this is a Find the Factors 1 to 14 puzzle, this pot of gold has some choice mathematical nuggets. For example, is 7 or 14 the common factor of 70 and 84? Don’t guess which one is the common factor for the puzzle. Use logic to eliminate one of those possibilities instead.

Likewise, both 6 and 9 are common factors of 18 and 54. And 4, 5, and 10 are all common factors of 20 and 40. Logic will narrow each possibility down to one possible factor!

Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

I like that you also need to find the common factor of 126 and 36. I noticed a pattern with those clues. The pattern is limited to the multiplication facts given below, but I think it is still a pretty cool pattern.

Here’s the pattern I saw for 126 and 36:

  • Since 9 is one of the factors, the sum of the digits of any of the products equals 9.
  • 1 + 2 = 3. The sum of the first two numbers of the product in the first column equals the first part of the product in the second column.
  • Obviously, both clues end in 6 so the last digit of their other factors will end with the same number, 4.

That should give you a good start in solving the puzzle!

Factors of 1609:

  • 1609 is a prime number.
  • Prime factorization: 1609 is prime.
  • 1609 has no exponents greater than 1 in its prime factorization, so √1609 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1609 has exactly 2 factors.
  • The factors of 1609 are outlined with their factor pair partners in the graphic below.

How do we know that 1609 is a prime number? If 1609 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1609. Since 1609 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1609 is a prime number.

More about the Number 1609:

1609² = 2588881. That’s a perfect square, followed by four 8’s, followed by a perfect square. OEIS.org reports that 1609² is the smallest perfect square with four 8’s in a row.

1609 is the sum of two squares:
40² + 3² = 1609.

1609 is the hypotenuse of a Pythagorean triple:
240-1591-1609, calculated from 2(40)(3), 40² – 3², 40² + 3².

Here’s another way we know that 1609 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 3² = 1609 with 40 and 3 having no common prime factors, 1609 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1609. Since 1609 is not divisible by 5, 13, 17, 29, or 37, we know that 1609 is a prime number.

 

1607 Shillelagh

Today’s Puzzle:

A Shillelagh is an Irish wooden walking stick. This Shillelagh is keeping with our Saint Patrick’s Day theme, but it is a Find the Factors 1 to 14 puzzle.  Brutal! It will be a whole lot less tricky for you to solve because I made it a level 3 puzzle: The logic needed to solve the puzzle is built in. Just start with the clue at the top of the puzzle and work your way down cell by cell until you have found all the factors. So crack on!

Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

Factors of 1607:

  • 1607 is a prime number.
  • Prime factorization: 1607 is prime.
  • 1607 has no exponents greater than 1 in its prime factorization, so √1607 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1607 has exactly 2 factors.
  • The factors of 1607 are outlined with their factor pair partners in the graphic below.

How do we know that 1607 is a prime number? If 1607 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1607. Since 1607 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1607 is a prime number.

More about the Number 1607:

1607 is the sum of two consecutive numbers:
803 + 804 = 1607.

1607 is also the difference of two consecutive numbers:
804² – 803² = 1607.

Did you notice what happened there? Try this next one:

1607² = 2582449.

1607²/2 = 1291224.5.

(1607-1291224-1291225) is a primitive Pythagorean triple.

Cool, isn’t it?

1601 and Level 6

Today’s Puzzle:

Remember to use logic for EVERY step when solving this puzzle. Guessing and checking will likely just frustrate you! It’s a level 6 puzzle, so it could be tricky.
Keep in mind:
1 and 2 are common factors of 6 and 8,
3 and 9 are common factors of 9 and 27,
4 & 8 are common factors of 32 and 8, and
4, 5, & 10 are common factors of 20 & 40.

As always, there is only one solution.

Factors of 1601:

  • 1601 is a prime number.
  • Prime factorization: 1601 is prime.
  • 1601 has no exponents greater than 1 in its prime factorization, so √1601 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1601 has exactly 2 factors.
  • The factors of 1601 are outlined with their factor pair partners in the graphic below.

How do we know that 1601 is a prime number? If 1601 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1601. Since 1601 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1601 is a prime number.

More about the Number 1601:

1601 is one more than a perfect square, so it is the sum of two squares:
40² + 1² = 1601.

1601 is the hypotenuse of a Pythagorean triple:
80-1599-1601, calculated from 2(40)(1), 40² – 1², 40² + 1².

Here’s another way we know that 1601 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 1² = 1601 with 40 and 1 having no common prime factors, 1601 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1601. Since 1601 is not divisible by 5, 13, 17, 29, or 37, we know that 1601 is a prime number.

1597 and Level 3

Today’s Puzzle:

You can solve this level 3 puzzle! Each number from 1 to 10 must appear in both the first column and the top row.

What is the greatest common factor of 24 and 56? Write that number above the column in which those clues appear. Write the corresponding factors in the first column. Next, starting with 72, write the factors of each clue going down the puzzle row by row until you have found all the factors.

Factors of 1597:

  • 1597 is a prime number.
  • Prime factorization: 1597 is prime.
  • 1597 has no exponents greater than 1 in its prime factorization, so √1597 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1597 has exactly 2 factors.
  • The factors of 1597 are outlined with their factor pair partners in the graphic below.

How do we know that 1597 is a prime number? If 1597 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1597. Since 1597 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1597 is a prime number.

More about the Number 1597:

1597 is the 17th Fibonacci number. It is also the only 4-digit Fibonacci prime.

1597 is the sum of two squares:
34² + 21² = 1597.

1597 is the hypotenuse of a Pythagorean triple:
715-1428-1597, calculated from 34² – 21², 2(34)(21), 34² + 21².

Here’s another way we know that 1597 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 21² = 1597 with 34 and 21 having no common prime factors, 1597 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1597. Since 1597 is not divisible by 5, 13, 17, 29, or 37, we know that 1597 is a prime number.

1579 It’s Inauguration Day!

Today’s Puzzle:

This star puzzle is one thing I’m doing to commemorate this historic day when Joe Biden is inaugurated as the 46th President of the United States and Kamala Harris is inaugurated as Vice President!  He will be the oldest person to become President, having previously served 36 years in the Senate and 8 years as Vice President, and she will be the first woman, the first African-American, and the first Asian-American Vice President. I wish them a beautiful day as they begin the hard work of uniting our country and finding solutions that benefit all of us.

The clues 10, 20, 30, and 40 have two common factors that might work for this mystery level puzzle. However, the other factors that go with one of those two choices will completely eliminate every possible factor pair for clue 4. That means you need to go with the other possibility.

Factors of 1579:

  • 1579 is a prime number.
  • Prime factorization: 1579 is prime.
  • 1579 has no exponents greater than 1 in its prime factorization, so √1579 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1579 has exactly 2 factors.
  • The factors of 1579 are outlined with their factor pair partners in the graphic below.

How do we know that 1579 is a prime number? If 1579 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1579. Since 1579 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1579 is a prime number.

More About the Number 1579:

1579 is the sum of two consecutive numbers:
789 + 790 = 1579.

1579 is also the difference of two consecutive squares:
790² – 789² = 1579.

1579, 915799, 99157999, 9991579999, and 999915799999 are all prime numbers! Thanks to OEIS.org for alerting me to that fabulous fact!

1567 Peppermint Stick

Today’s Puzzle:

Our mystery level puzzle looks like a sweet stick of Christmas candy. Will solving it be sweet or will it be sticky? You’ll have to try it yourself to know.

Factors of 1567:

  • 1567 is a prime number.
  • Prime factorization: 1567 is prime.
  • 1567 has no exponents greater than 1 in its prime factorization, so √1567 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1567 has exactly 2 factors.
  • The factors of 1567 are outlined with their factor pair partners in the graphic below.

How do we know that 1567 is a prime number? If 1567 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1567. Since 1567 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1567 is a prime number.

More about the Number 1567:

1567 is the sum of two consecutive numbers:
783 + 784 = 1567.

1567 is also the difference of two consecutive squares:
784² – 783² = 1567.

1559 We Need a Little Christmas Now

Today’s Puzzle:

2020 reminds me of another difficult year, 2001. In December of that year, Angela Lansbury sang We Need a Little Christmas Now with the Tabernacle Choir at Temple Square. I was able to watch the concert on television, and I remember the feeling the music brought me. What a wonderful gift music is! Yes, in 2020, we need a little Christmas now!

This level 2 puzzle brings a little Christmas now. Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions as a type of multiplication table. I’m pretty sure you can figure it out!

Factors of 1559:

  • 1559 is a prime number.
  • Prime factorization: 1559 is prime.
  • 1559 has no exponents greater than 1 in its prime factorization, so √1559 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1559 has exactly 2 factors.
  • The factors of 1559 are outlined with their factor pair partners in the graphic below.

How do we know that 1559 is a prime number? If 1559 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1559. Since 1559 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1559 is a prime number.

More about the number 1559:

1559 is the sum of two consecutive numbers:
780 + 779 = 1559.

1559 is also the difference of two consecutive square numbers:
780² – 779² = 1559.

(Yes, I know, any odd whole number can make a similar claim.)