1616 Centering the Pendulum

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.

1612 Celtic Knot

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?

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.

1602 Mystery

Today’s Puzzle:

Is the logic needed to solve this puzzle simple or complicated? That question is part of the mystery!

Factors of 1602:

  • 1602 is a composite number.
  • Prime factorization: 1602 = 2 × 3 × 3 × 89, which can be written 1602 = 2 × 3² × 89.
  • 1602 has at least one exponent greater than 1 in its prime factorization so √1602 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1602 = (√9)(√178) = 3√178.
  • 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 1602 has exactly 12 factors.
  • The factors of 1602 are outlined with their factor pair partners in the graphic below.

More about the Number 1602:

How is 1602 the sum of two squares?
39² + 9² = 1602.

How is 1602 the hypotenuse of a Pythagorean triple?
702-1440-1602, calculated from 2(39)(9), 39² – 9², 39² + 9².
That triple is also 9 times (78-160-178).

1592 One More Valentine

Today’s Puzzle:

I made this mystery level puzzle into one more valentine. Love can seem tricky sometimes, but I hope you enjoy working on it.

Factors of 1592:

  • 1592 is a composite number.
  • Prime factorization: 1592 = 2 × 2 × 2 × 199, which can be written 1592 = 2³ × 199.
  • 1592 has at least one exponent greater than 1 in its prime factorization so √1592 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1592 = (√4)(√398) = 2√398.
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1592 has exactly 8 factors.
  • The factors of 1592 are outlined with their factor pairs in the graphic below.

More about the Number 1592:

1599 is the difference of two squares two different ways:
399² – 397² = 1592, and
201² – 197² = 1592.

1583 The Logic for This Mystery Level Puzzle

Today’s Puzzle:

The logic for this mystery level puzzle starts out simple, but gets more complicated as you go along. Will you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues?

Factors of 1583:

  • 1583 is a prime number.
  • Prime factorization: 1583 is prime.
  • 1583 has no exponents greater than 1 in its prime factorization, so √1583 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1583 has exactly 2 factors.
  • The factors of 1583 are outlined with their factor pair partners in the graphic below.

How do we know that 1583 is a prime number? If 1583 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1583. Since 1583 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1583 is a prime number.

More about the Number 1583:

1583 is the sum of two consecutive numbers:
791 + 792 = 1583.

1583 is also the difference of two consecutive squares:
792² – 791² = 1583.

What do you notice about the consecutive numbers in those two facts?

 

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!

1571 Candy Cane

Today’s Puzzle:

The shepherds’ crooks from that first Christmas night have become the sweet candy canes we often see on today’s Christmas trees. Can you find the factors from 1 to 12 that will make this mystery level puzzle function like a multiplication table? Remember to use logic to find the factors.

Factors of 1571:

  • 1571 is a prime number.
  • Prime factorization: 1571 is prime.
  • 1571 has no exponents greater than 1 in its prime factorization, so √1571 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1571 has exactly 2 factors.
  • The factors of 1571 are outlined with their factor pair partners in the graphic below.

How do we know that 1571 is a prime number? If 1571 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1571. Since 1571 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1571 is a prime number.

More about the Number 1571:

1571 is the sum of two consecutive numbers:
785 + 786 = 1571.

1571 is also the difference of two squares:
786² – 785² = 1571.

Do you see the relationship between those two facts?

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.

1565 Stable with Manger

Today’s Puzzle:

This mystery level puzzle reminds me of the manger in the stable that first Christmas night.

How difficult will the puzzle be to solve? That is part of the mystery. You will have to try it for yourself to find out.

Factors of 1565:

  • 1565 is a composite number.
  • Prime factorization: 1565 = 5 × 313.
  • 1565 has no exponents greater than 1 in its prime factorization, so √1565 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 1565 has exactly 4 factors.
  • The factors of 1565 are outlined with their factor pair partners in the graphic below.

More about the Number 1565:

1565 is the sum of two squares in two different ways:
37² + 14² = 1565, and
38² + 11² = 1565.

125-1560-1565, which is 5 times (25-312-313),
836-1323-1565, calculated from 2(38)(11), 38² – 11², 38² + 11²,
939-1252-1565, which is (3-4-5) times 313, and
1036-1173-1565, calculated from 2(37)(14), 37² – 14², 37² + 14².