1616 Centering the Pendulum

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Today’s Puzzle:

Centering the Pendulum is Bill Davidson’s podcast about educators and how they inspire students to learn mathematics. All of his podcasts are wonderful and more than worth the 15 minutes or so needed to listen to each one.  I am quite honored that Find the Factors is the subject of his fourth podcast, and you can listen to it here.

To mark this occasion, I’ve made a mystery level puzzle that resembles a swinging pendulum that hopefully is centered! The puzzle might or might not be a little tricky, but just use logic every step of the way, and you should be fine.

Using logic, write the numbers from 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. Have fun!

Factors of 1616:

  • 1616 is a composite number.
  • Prime factorization: 1616 = 2 × 2 × 2 × 2 × 101, which can be written 1616 = 2⁴ × 101.
  • 1616 has at least one exponent greater than 1 in its prime factorization so √1616 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1616 = (√16)(√101) = 4√101.
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1616 has exactly 10 factors.
  • The factors of 1616 are outlined with their factor pair partners in the graphic below.

More about the Number 1616:

1616 is the sum of two squares:
40² + 4² = 1616.

1616 is also the hypotenuse of a Pythagorean triple:
320-1584-1616, which is 16 times (20-99-101)
and can be calculated from 2(40)(4), 40² – 4², 40² + 4².

1616 is also the difference of two squares in three ways:
405² – 403² = 1616,
204² – 200² = 1616, and
105² – 97² = 1616.

1615 Should Today Be Root Ten Day?

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Today’s Puzzle:

In a recent post, I compared π or (3.14…) with √10 or (3.16…). Steve Morris lives in England where today’s date is written 16-3, not 3-16. He jokingly commented, “So I guess Tuesday (16 March) should be Root Ten Day!” Seriously, day-month-year makes more sense as a writing convention than month-day-year.

Should today be Root Ten Day?
14 March has long been embraced as pi day in the United States, but should 16 March also be a quasi-holiday where kids eat roots like ten French fries or ten carrot sticks?

I remember one of my college professors telling his class that
√2 is about 1.4, and Valentines day is February 14,
√3 is about 1.7, and Saint Patrick’s day is March 17.

To which we could add
√1 is 1, and New Year’s Day is January 1, and
√10 is about 3.1, and Halloween is October 31. (I realize there is a rounding issue with that one.)

Oops. That could be said about all the fake holidays I’ve listed above.

And here’s a more serious thought:

Well, however you want to remember what √10 is or not, I decided to make today’s puzzle look like a square root sign for the fun of it. Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1615:

  • 1615 is a composite number.
  • Prime factorization: 1615 = 5 × 17 × 19.
  • 1615 has no exponents greater than 1 in its prime factorization, so √1615 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 1615 has exactly 8 factors.
  • The factors of 1615 are outlined with their factor pair partners in the graphic below.

More about the Number 1615:

1615 is the hypotenuse of FOUR Pythagorean triples:
247-1596-1615, which is 19 times (13-84-85),
684-1463-1615, which is 19 times (36-77-85),
760-1425-1615, which is (8-15-17) times 95, and
969-1292-1615, which is (3-4-5) times 323.

1614 and Level 1

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Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1614:

1614 is even so it’s divisible by 2.

Here’s the weird thought process I used to determine its divisibility by 3:
161514 is made from three consecutive numbers, so it’s divisible by 3. We know that 15 is divisible by 3, so we can remove it. That means 1614 is divisible by 3.

Since 1614 is divisible by both 2 and 3, it is divisible by 6.

  • 1614 is a composite number.
  • Prime factorization: 1614 = 2 × 3 × 269.
  • 1614 has no exponents greater than 1 in its prime factorization, so √1614 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 1614 has exactly 8 factors.
  • The factors of 1614 are outlined with their factor pair partners in the graphic below.

More about the Number 1614:

1614 is the hypotenuse of a Pythagorean triple:
414-1560-1614, which is 6 times (69-260-269).

1614 is also a leg in two Pythagorean triples because
2(807)(1) = 1614, and
2(269)(3) = 1614.

You can calculate those Pythagorean triples by letting a be the first number in parenthesis for each of those equations, and b be the second number in parenthesis. Then substitute those values in the three expressions below, and you will have some Pythagorean triples!
2(a)(b), a² – b², a² + b².

1613 Comparing π and √10

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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.

 

1612 Celtic Knot

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Today’s Puzzle:

Many Celtic knots have no beginning and no end, so they are mathematically interesting. This one is like a Trinity Knot, and it doubles as a mystery-level puzzle. Solving it might be a little tricky, but it will still be lots of fun.

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

Factors of 1612:

  • 1612 is a composite number.
  • Prime factorization: 1612 = 2 × 2 × 13 × 31, which can be written 1612 = 2² × 13 × 31.
  • 1612 has at least one exponent greater than 1 in its prime factorization so √1612 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1612 = (√4)(√403) = 2√403.
  • 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 1612 has exactly 12 factors.
  • The factors of 1612 are outlined with their factor pair partners in the graphic below.

More about the Number 1612:

1612 can be written as a palindromic product in two different ways:
2 × 13 × 31 × 2,
26 × 62.

1612 is the hypotenuse of a Pythagorean triple:
620- 1488-1612, which is (5-12-13) times 124.

1612 can be written as the difference of two squares in two ways:
404² – 402² = 1612, and
44² – 18² = 1612.

 

 

1611 A Little Blarney?

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Today’s Puzzle:

Many travelers to Ireland hang upsidedown to kiss the Blarney Stone at the top of Blarney castle. There is even a Sherlock Holmes mystery about someone who appears to have fallen to his death while trying to kiss the Blarney Stone. If I said this mystery level puzzle represents the Blarney Stone, would that just be a bunch of blarney?

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

Factors of 1611:

  • 1611 is a composite number.
  • Prime factorization: 1611 = 3 × 3 × 179, which can be written 1611 = 3² × 179.
  • 1611 has at least one exponent greater than 1 in its prime factorization so √1611 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1611 = (√9)(√179) = 3√179.
  • 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 1611 has exactly 6 factors.
  • The factors of 1611 are outlined with their factor pair partners in the graphic below.

More about the number 1611:

1611 is the difference of two squares in three different ways:
806² – 805² = 1611,
270² – 267² = 1611, and
94² – 85² = 1611.

1610 Four-Leaf Clovers

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Today’s Puzzle:

Sometimes four-leaf clovers are associated with Saint Patrick’s Day. Four-leaf clovers are supposed to be lucky, but you might not feel so lucky as you work on solving this puzzle. I assure you, there is a logical way to proceed on each step!

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

Factors of 1610:

  • 1610 is a composite number.
  • Prime factorization: 1610 = 2 × 5 × 7 × 23.
  • 1610 has no exponents greater than 1 in its prime factorization, so √1610 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1610 has exactly 16 factors.
  • The factors of 1610 are outlined with their factor pair partners in the graphic below.

More about the Number 1610:

1610 is the hypotenuse of a Pythagorean triple:
966-1288-1610, which is (3-4-5) times 322.

1610 is not the sum of two squares or the difference of two squares. 1610 is a leg in some Pythagorean triples because
2(805)(1) = 1610,
2(161)(5) = 1610,
2(115)(7) = 1610, and
2(35)(23) = 1610.

You can calculate those Pythagorean triples by letting a be the first number in parenthesis for each of those equations, and b be the second number in parenthesis. Then substitute those values in the three expressions below, and you will have some Pythagorean triples!
2(a)(b), a² – b², a² + b².

1609 Pot of Gold

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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.

 

1608 Rainbow

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Today’s Puzzle:

We often think of rainbows around Saint Patrick’s Day. Here is a rainbow puzzle for you to solve. It won’t be all that easy even if I tell you that
13 × 14 = 182,
12 × 13 = 156, and
8 ×  14 = 112.

Good luck!

If you’d like to print the puzzle but not use so much ink, here’s a puzzle with all the same clues:

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

Factor Rainbow for 1608:

The number 1608 has enough factors to make an impressive factor rainbow:

Factors of 1608:

  • 1608 is a composite number.
  • Prime factorization: 1608 = 2 × 2 × 2 × 3 × 67, which can be written 1608 = 2³ × 3 × 67.
  • 1608 has at least one exponent greater than 1 in its prime factorization so √1608 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1608 = (√4)(√402) = 2√402.
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1608 has exactly 16 factors.
  • The factors of 1608 are outlined with their factor pair partners in the graphic below.

More about the Number 1608:

1608 is the difference of two squares in four different ways:
403² – 401² = 1608,
203² – 199² = 1608,
137² – 131² = 1608, and
73² – 61² = 1608.

1607 Shillelagh

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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?