1581 One Teardrop among Millions

Today’s Puzzle:

Earlier this week the United States surpassed 400,000 deaths from the novel coronavirus. Over 2,000,000 people have died from the virus worldwide. Most of these people died in isolation away from loved ones. Most of the millions of tears shed have also been in isolation. New and more contagious strains of the virus have arisen in various parts of the world threatening to make the death toll worse and the number of tears to grow exponentially. Each of us can and must do our part to flatten the curve and thus restrict the number of deaths and the number of tears.

Factors of 1581:

  • 1581 is a composite number.
  • Prime factorization: 1581 = 3 × 17 × 31.
  • 1581 has no exponents greater than 1 in its prime factorization, so √1581 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 1581 has exactly 8 factors.
  • The factors of 1581 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1581:

1581 is the hypotenuse of a Pythagorean triple:
744-1395-1581, which is (8-15-17) times 93.

1580 Use Logic to Solve This Puzzle

Today’s Puzzle:

Where should you write the numbers from 1 to 10 in both the first column and the top row to make this puzzle function like a multiplication table? Study it until you find a logical place to start and continue to use logic until you’ve completed the puzzle.

Factors of 1580:

  • 1580 is a composite number.
  • Prime factorization: 1580 = 2 × 2 × 5 × 79, which can be written 1580 = 2² × 5 ×
  • 1580 has at least one exponent greater than 1 in its prime factorization so √1580 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1580 = (√4)(√395) = 2√
  • 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 1580 has exactly 12 factors.
  • The factors of 1580 are outlined with their factor pair partners in the graphic below.

More about the Number 1580:

1580 is the difference of two squares in two different ways:
396² – 394² = 1580, and
84² – 74² = 1580.

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!

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.

1577 Crossing a Bridge

Today’s Puzzle:

Merriam-Webster defines a bridge as “a structure carrying a pathway or roadway over a depression or obstacle (such as a river)”. Bridges have been built over wide rivers, but what type of bridge is needed when the obstacle is prejudice, hate, or greed?

Today is Martin Luther King Jr. Day in the United States. He tirelessly worked on such bridges. The movie, Selma, portrays much of what he and his companions did to cross a certain bridge in Alabama.  Although ALL of his protests were peaceful, some who followed him were brutally killed. Dr. King himself was assassinated when he was just 39 years old.

Today bridges still need to be built. Education can be an important bridge-builder. If you have Netflix, type “Martin Luther King” into the search, and several eye-opening movies and documentaries will appear. I highly recommend 13th, a documentary that shows how the same amendment to the constitution that abolished slavery has a loophole that essentially allows slavery to continue to this day. I was appalled.

The puzzle below is easy to solve. How to build bridges that make the world better for EVERYONE is a far more important and difficult puzzle that each of us needs to ponder to get closer to a solution.

Factors of 1577:

  • 1577 is a composite number.
  • Prime factorization: 1577 = 19 × 83.
  • 1577 has no exponents greater than 1 in its prime factorization, so √1577 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 1577 has exactly 4 factors.
  • The factors of 1577 are outlined with their factor pair partners in the graphic below.

More about the Number 1577:

1577 is the difference of two squares two different ways:
789² – 788² = 1577, and
51² – 32² = 1577.

1577 is the sum of the 2 numbers from 788 to 789.
1577 is the sum of the 19 numbers from 74 to 92. (83 is in the exact middle of that list of numbers.)
1577 is the sum of the 38 numbers from 23 to 60.

1576 Why Do the Digits of Every Clue in This Puzzle Add up to Nine?

Today’s Puzzle:

The digits of each clue in this puzzle add up to nine. Relatively few Find the Factors puzzles can make that claim. Why can this one make it?

Can you write all the numbers from 1 to 10 so that this puzzle will function like a multiplication table?

Factors of 1576:

1576 is divisible by a few powers of two, namely 1, 2, 4, and 8. Here is something for you to think about:

1576 is a whole number so it is divisible by 1.
6 is even so 1576 is divisible by 2.
7 is odd and 6 is not divisible by 4, so 1576 is divisible by 4.
5 is odd and 76 is not divisible by 8, so 1576 is divisible by 8.
1 is odd and 576 is divisible by 16, so 1576 is not divisible by 16.
(Each of those statements was also dependent on the statement before it.)

  • 1576 is a composite number.
  • Prime factorization: 1576 = 2 × 2 × 2 × 197, which can be written 1576 = 2³ × 197
  • 1576 has at least one exponent greater than 1 in its prime factorization so √1576 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1576 = (√4)(√394) = 2√394
  • 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 1576 has exactly 8 factors.
  • The factors of 1576 are outlined with their factor pair partners in the graphic below.

More about the Number 1576:

1576 is the sum of two squares:
30² + 26² = 1576.

1576 is the hypotenuse of a Pythagorean triple:
224-1560-1576, calculated from 30² – 26², 2(30)(26), 30² + 26².
It is also 8 times (28-195-197).

1576 is also the difference of two squares in two different ways:
395² – 393² = 1576 and
199² – 195² = 1576.

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?

1570 How to Solve the Dollar Tree Ball Puzzle

Today’s Puzzle:

Perhaps you or one of your children will find the Dollar Tree Ball Puzzle in your stocking this year. Such was the case in my family.

A little bit of tugging here and there will help you pull apart this Dollar Tree puzzle ball, but how do you get it back together? After several failed attempts, you might turn to the internet for help. There are several videos showing how to put it back together, but they can be difficult to follow. The colors of the puzzle pieces might not match your ball. Thus, instructions on what to do with the yellow or green piece are not as helpful. People on the videos keep turning the ball so that makes the differences in color even more confusing and requires the videos to be played back over and over again.

Instead of posting yet another video, I decided to post still shots of each step with the Dollar Tree Puzzle Ball maintaining . The colors of your ball may not match the colors here, so match the indentations on your puzzle pieces instead of the colors. Turn your puzzle pieces so the indentations are in the same position as the indentations in the photos.

First 2 pieces: Find the two pieces shaped like these:

Connect them so that together they look like an angel.  Third piece: There are actually two puzzle pieces that are shaped the same. Select one of them.

Put your selected piece below the angel wings on the puzzle. Fourth piece: select the next piece that is shaped like the blue one below. See how the notch will fit over the center of the angel wings perfectly?

Attach it to the puzzle as shown. Notice the two holes in the middle of the puzzle. The one on the left is a rectangle. The one on the right is a square.

Fifth piece: Put the next piece in on the rectangle side with the solid part down. Push it in all the way to the middle.

Sixth piece: Put the last piece in. This is the piece that was identical to the third piece you put in the puzzle.

Push it in all the way to the middle.

Finally, push the bottom piece up to lock the puzzle into place.

Factors of 1570:

This is my 1570th post. Separating the number 1570 into puzzle pieces might seem difficult, but it only has three prime puzzle pieces: 2, 5, and 157. That last one might seem like it should be broken into more pieces, but it can’t be. Putting the puzzle pieces back together is much easier. Just multiply them all together to get 1570.

  • 1570 is a composite number.
  • Prime factorization: 1570 = 2 × 5 × 157.
  • 1570 has no exponents greater than 1 in its prime factorization, so √1570 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 1570 has exactly 8 factors.
  • The factors of 1570 are outlined with their factor pair partners in the graphic below.

More about the Number 1570:

1570 is the sum of two squares in two different ways:
29² + 27² = 1570, and
39² + 7² = 1570.

1570 is the hypotenuse of four Pythagorean triples:
112-1566-1570, calculated from 29² – 27², 2(29)(27), 29² + 27²,
546-1472-1570, calculated from 2(39)(7), 39² – 7², 39² + 7²
850-1320-1570, which is 10 times (85-132-157), and
942-1256-1570, which is (3-4-5) times 314.

 

1568 A Challenging Christmas Tree

Today’s Puzzle:

Can you write the numbers from 1 to 10 in each of the boldly outlined columns and rows so that each quadrant of this Christmas tree puzzle behaves like a multiplication table? Remember to use logic and not guessing and checking.

Here is the same puzzle that won’t use so much ink to print:

1568 Factor Tree:

Here is one of several possible factor trees for 1568:

Factors of 1568:

  • 1568 is a composite number.
  • Prime factorization: 1568 = 2 × 2 × 2 × 2 × 2 × 7 × 7, which can be written 1568 = 2⁵ × 7²
  • 1568 has at least one exponent greater than 1 in its prime factorization so √1568 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1568 = (√784)(√2) = 28√2
  • The exponents in the prime factorization are 5 and 2. Adding one to each exponent and multiplying we get (5 + 1)(2 + 1) = 6 × 3 = 18. Therefore 1568 has exactly 18 factors.
  • The factors of 1568 are outlined with their factor pair partners in the graphic below.

More about the Number 1568:

1568 is the difference of two squares SIX different ways:
393² – 391² = 1568,
198² – 194² = 1568,
102² – 94² = 1568,
63² – 49² = 1568,
57² – 41² = 1568, and
42² – 14² = 1568.

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.