1675 You CAN Solve This Level 3 Puzzle!

Great news for math enthusiasts, students, and teachers everywhere! The #148 Playful Math Education Carnival was published today at Math Book Magic!

Today’s puzzle:

I am confident that you can solve this level 3 puzzle! Here’s how: Using only the numbers from 1 to 10, write the factors of 27 and 6 in the appropriate cells. Next, write 18’s factors. Then, since this is a level 3 puzzle, write the factors of 30, 50, 40, 36, 14, 56, and 8, in that order, until all the numbers from 1 to 10 appear in the first column as well as in the top row. As always, there is only one solution.

What did I tell you? You could solve it!

Factors of 1675:

  • 1675 is a composite number.
  • Prime factorization: 1675 = 5 × 5 × 67, which can be written 1675 = 5² × 67.
  • 1675 has at least one exponent greater than 1 in its prime factorization so √1675 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1675 = (√25)(√67) = 5√67.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1675 has exactly 6 factors.
  • The factors of 1675 are outlined with their factor pair partners in the graphic below.

More About the Number 1675:

Did you notice that 1675 = 5 × 67 × 5?

How many quarters are in $16.75?
Well, 17 × 4 quarters = 68 quarters = $17.00.
Subtracting one quarter from both sides of the equation, we get
67 quarters = $16.75.

1675 is the hypotenuse of TWO Pythagorean triples:
469-1608-1675, which is (7-24-25) times 67, and
1005-1340-1675, which is (3-4-5) times 335.

1675 is the difference of two squares in THREE different ways:
838² – 837² =  1675,
170² – 165² =  1675, and
46² – 21²  =  1675.

1666 Demystifying a Tricky Puzzle

Today’s Puzzle:

This puzzle isn’t as tricky as it could be simply because I arranged its clues into a level 3 puzzle. That means that after you write the factors of 99 and 18 in the appropriate boxes in the first column and top row, you work your way down the puzzle clue by clue in the order they appear. Still, you will have to think about what to do with the 70, but I think you can handle it!

Factors of 1666:

  • 1666 is a composite number.
  • Prime factorization: 1666 = 2 × 7 × 7 × 17, which can be written 1666 = 2 × 7² × 17.
  • 1666 has at least one exponent greater than 1 in its prime factorization so √1666 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1666 = (√49)(√34) = 7√34.
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1666 has exactly 12 factors.
  • The factors of 1666 are outlined with their factor pair partners in the graphic below.

More About the Number 1666:

1666 is the sum of two squares because ALL of its odd prime factors either leave a remainder of 1 when divided by 4 OR have an even exponent:
1666 = 2 × 7² × 17,
17÷4 = 4 R1, The exponent on 7² is even.
What are the two squares?
35² +  21² = 1666.

1666 is the hypotenuse of a Pythagorean triple :
784-1470-1666 which is (8-15-17) times 98 and
can also be calculated from 35² –  21², 2(35)(21), 35² +  21².

 

1652 Start at the Top and Work Your Way Down to the Bottom

Today’s Puzzle:

This is a level 3 puzzle so the clues are given in a logical order starting from the top of the puzzle. Begin by writing the factors of 20 and 32 in the appropriate cells. Then write the rest of the numbers so that both the first column and the top row have all the numbers from 1 to 10, and the written numbers are the factors of the given clues.

Factors of 1652:

  • 1652 is a composite number.
  • Prime factorization: 1652 = 2 × 2 × 7 × 59, which can be written 1652 = 2² × 7 × 59.
  • 1652 has at least one exponent greater than 1 in its prime factorization so √1652 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1652 = (√4)(√413) = 2√413.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1652 has exactly 12 factors.
  • The factors of 1652 are outlined with their factor pair partners in the graphic below.

More About the Number 1652:

1652 is the difference of two squares two different ways:
414² – 412² = 1652 and
66² – 52² = 1652.

1641 and Level 3

Today’s Puzzle:

This is a level 3 puzzle so the clues have been placed so that you can know what order you should use the clues. Place the factors of 90 and 30 in the appropriate cells, then work your way down the puzzle cell by cell filling in the factors of the clues as you go.

Factors of 1641:

1 + 4 + 1 = 6, so 1641 is divisible by 3. (It isn’t necessary to include multiples of 3 in the sum to determine divisibility by 3.)

  • 1641 is a composite number.
  • Prime factorization: 1641 = 3 × 547.
  • 1641 has no exponents greater than 1 in its prime factorization, so √1641 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1641 has exactly 4 factors.
  • The factors of 1641 are outlined with their factor pair partners in the graphic below.

More About the Number 1641:

1641 is the difference of two squares in two different ways:
821² – 820² = 1641, and
275² – 272² = 1641.

From OEIS.org we learn that the number formed from 1²6²4²1² is a perfect square:
Recall that 1² = 1; 6² = 36; 4² = 16; and 1² = 1. Those squares form the number, 136161.
Sure enough, 136161 = 369².

1630 and Level 3

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so those numbers and the given clues make the puzzle function like a multiplication table. Because this is a level 3 puzzle, first write the factors for 72 and 90. Then work your way down the puzzle row by row until you have found all the factors.

Factors of 1630:

  • 1630 is a composite number.
  • Prime factorization: 1630 = 2 × 5 × 163.
  • 1630 has no exponents greater than 1 in its prime factorization, so √1630 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1630 has exactly 8 factors.
  • The factors of 1630 are outlined with their factor pair partners in the graphic below.

More about the Number 1630:

1630 is the hypotenuse of a Pythagorean triple:
978-1304-1630, which is (3-4-5) times 326.

1618 Math Happens in the Austin Chronicle

Math Happens!

Several years before I started blogging, I tried to get my puzzles in newspapers, but the publishers of those newspapers just ignored them. Because of that, it is even sweeter to me that Math Happens put one of them in the Austin Chronicle! You can see it in the newspaper on page 25 of this pdf or in this cool page-turning e-edition. Math Happens in many different ways as you can see in their blog post from February 5. You can also look for Math Happens on a page in the middle of each of these  2020 issues or 2021 issues of the Austin Chronicle newspaper online.

Math Happens also in the Orange Leader, and they would love to also be in your local community newspaper.

You can have your local newspaper contact them through Twitter!

Today’s Puzzle:

Spring happens in just a few days! Today’s puzzle represents grasses blowing in a spring wind, readily anticipating the hiding of Easter eggs. It’s a level 3 puzzle, so start by finding the factors of the clue at the top of the puzzle (and the clue that goes with it), and work your way down cell by cell until you have written all the numbers from 1 to 12 in both the factor column and the factor row. You can do this!

Factors of 1618:

  • 1618 is a composite number.
  • Prime factorization: 1618 = 2 × 809.
  • 1618 has no exponents greater than 1 in its prime factorization, so √1618 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1618 has exactly 4 factors.
  • The factors of 1618 are outlined with their factor pair partners in the graphic below.

More about the Number 1618:

1618 = 2 × 809, and 2809 is a perfect square. Thank you OEIS.org for that fun fact!

1618 is the sum of two squares:
33² + 23² = 1618.

1618 is the hypotenuse of a Pythagorean triple:
560-1518-1618, calculated from 2(33)(23), 33² – 23², 33² + 23².
It is also 2 times (280-759-809).

1618 is the 22nd centered heptagonal number because it is one more than seven times the 21st triangular number:
7(21)(22)/2 + 1 = 1618.

1618 has exactly four factors. The last number with exactly four factors was 1603. That’s the biggest gap so far between two numbers with exactly four factors!
(It will be interesting to see who will win the horse race for the current set of 100 numbers. So far, the horses for 2 factors and 8 factors are each running twice as fast as the horse for 4 factors, and 1619 will be a prime number, giving 2 factors the lead!)

A lot of math is happening with this 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?

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.

1588 Cupid’s Arrow

Today’s Puzzle:

Keeping with our Valentine’s theme, today’s level 3 puzzle looks like Cupid’s Arrow. Start with the clues at the top of the arrow, write in their factors, and work your way down the puzzle, cell by cell, writing in factors as you go. Before long, you will be smitten with this puzzle!

Factors of 1588:

  • 1588 is a composite number.
  • Prime factorization: 1588 = 2 × 2 × 397, which can be written 1588 = 2² × 397.
  • 1588 has at least one exponent greater than 1 in its prime factorization so √1588 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1588 = (√4)(√397) = 2√397.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1588 has exactly 6 factors.
  • The factors of 1588 are outlined with their factor pair partners in the graphic below.

More about the number 1588:

1588 is the difference of two squares:
398² – 396² = 1588.

1588 is also the sum of two squares:
38² + 12² = 1588.

1588 is the hypotenuse of a Pythagorean triple:
912-1300-1588, calculated from 2(38)(12), 38² – 12², 38² + 12².
It is also 4 times (228-325-397).

1578 The Logic Needed to Solve This Puzzle is Straight Forward

Today’s Puzzle:

What is the only common factor of 36 and 9 that will use only numbers from 1 to 10 in the first column? Answer that question, put the factors in their appropriate cells, and then go straight down the puzzle row by row, filling in factors as you go. In no time at all, you will have solved this level 3 puzzle!

Factors of 1578:

1 + 5 + 7 + 8 = 21, a number divisible by 3, so even number 1578 is divisible by 6.

  • 1578 is a composite number.
  • Prime factorization: 1578 = 2 × 3 × 263.
  • 1578 has no exponents greater than 1 in its prime factorization, so √1578 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1578 has exactly 8 factors.
  • The factors of 1578 are outlined with their factor pair partners in the graphic below.

More about the Number 1578:

1578 is a leg in two Pythagorean triples:
1578-622520-622522, calculated from 2(789)(1), 789² – 1², 789² – 1², and
1578-69160-69178, calculated from 2(263)(3), 263² – 3², 263² – 3².