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?

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.