Puzzle

1587 XOXOXO Hugs and Kisses OXOXOX

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Today’s Puzzle:

Hugs are often abbreviated as “O” in letters and notes, and kisses are abbreviated as “X”. This puzzle can be a Valentine’s card and will send hugs and kisses to whomever you give it.

Can you 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?

Factors of 1587:

1587 is divisible by 3 because 1 + 5 + 8 + 7 = 21, a number divisible by 3.
1587 ÷ 3 = 529. Perhaps you have memorized some perfect squares and remember that 23² = 529. (If your parents invest in a 529 college plan for your education, you are more likely to have a college degree by the time you are 23. That’s how I remember it!)

  • 1587 is a composite number.
  • Prime factorization: 1587 = 3 × 23 × 23, which can be written 1587 = 3 × 23².
  • 1587 has at least one exponent greater than 1 in its prime factorization so √1587 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1587 = (√529)(√3) = 23√3.
  • The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 1587 has exactly 6 factors.
  • The factors of 1587 are outlined with their factor pair partners in the graphic below.

More about the Number 1587:

1587 is the difference of two squares three different ways:
794² – 793² = 1587,
266² – 263² = 1587, and
46² – 23² = 1587.

 

1586 and a Level One Puzzle

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Today’s Puzzle:

Write the number from 1 to 12 in both the first column and the top row of the puzzle so that the given clues are the products of the numbers you write.

Factors of 1586:

  • 1586 is a composite number.
  • Prime factorization: 1586 = 2 × 13 × 61.
  • 1586 has no exponents greater than 1 in its prime factorization, so √1586 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 1586 has exactly 8 factors.
  • The factors of 1586 are outlined with their factor pair partners in the graphic below.

More about the Number 1586:

1586 is the sum of two squares in two different ways:
35² + 19² = 1586, and
31² + 25² = 1586.

1586 is the hypotenuse of FOUR Pythagorean triples:
286-1560-1586, which is 26 times (11-60-61),
336-1550-1586, calculated from 31² – 25², 2(31)(25), 31² + 25²,
610-1464-1586, which is (5-12-13) times 122, and
864-1330-1586, calculated from 31² – 25², 2(31)(25), 31² + 25².

1584 You Will L♥ve This Challenge Puzzle!

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Today’s Puzzle:

Here’s a Valentine’s challenge for you: Can you write the numbers from 1 to 10 in each of the four areas of the puzzle that touch the box with the X so that the numbers you write make the puzzle become four multiplication tables? It won’t be easy, but remember to use logic. If you succeed, you will absolutely l♥ve the puzzle!

1584 Factor Cake:

1584 obviously is divisible by 2. It is divisible by 4 because 84 is divisible by 4.
It is divisible by 8 because 84 is not divisible by 8 and 5 is odd.
1584 is divisible by 3 and by 9 because 1 + 5 + 8 + 4 = 18, a number divisible by 3 and by 9.
1584 is divisible by 11 because 1 – 5 + 8 – 4 = 0.

Thus, 1584 makes a very tall and very nice factor cake with a couple of candles on top!

Factors of 1584:

  • 1584 is a composite number.
  • Prime factorization: 1584 = 2 × 2 × 2 × 2 × 3 × 3 × 11, which can be written 1584 = 2⁴ × 3² × 11.
  • 1584 has at least one exponent greater than 1 in its prime factorization so √1584 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1584 = (√144)(√11) = 12√11.
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each exponent and multiplying we get (4 + 1)(2 + 1)(1 + 1) = 5 × 3 × 2 = 30. Therefore 1584 has exactly 30 factors.
  • The factors of 1584 are outlined with their factor pair partners in the graphic below.

More about the Number 1584:

1584 is the difference of two squares NINE different ways:
397² – 395² = 1584,
200² – 196² = 1584,
135² – 129² = 1584,
103² – 95² = 1584,
72² – 60² = 1584,
53² – 35² = 1584,
47² – 25² = 1584,
45² – 21² = 1584, and
40² – 4² = 1584.
Yes, we are only 16 numbers away from 1600, the next perfect square!

1584 is in this cool pattern:

 

1583 The Logic for This Mystery Level Puzzle

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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?

 

1582 Half the Time You’ll Be Thinking about Multiples instead of Factors

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Today’s Puzzle:

In the level 6 puzzle, the possible common factors of 40 and 20 are 4, 5, and 10 while the possible common factors of 24 and 12 are 3, 4, and 6.

Don’t worry about which common factor to choose to start the puzzle. Instead, think about multiples. One of the numbers from 1 to 10 will only be able to go in one place in the top row because that’s the only place one of its multiples is in the column below it. That same number will also have only one place it can go in the first column.

You will likely look for lone multiples of particular numbers a total of five times as you solve this puzzle. Good luck!

Factors of 1582:

  • 1582 is a composite number.
  • Prime factorization: 1582 = 2 × 7 × 113.
  • 1582 has no exponents greater than 1 in its prime factorization, so √1582 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 1582 has exactly 8 factors.
  • The factors of 1582 are outlined with their factor pair partners in the graphic below.

More about the Number 1582:

1582 is the hypotenuse of a Pythagorean triple:
210-1568-1582, which is 14 times (15-112-113).

1582 is the sum of the four numbers from 394 to 397.

1582 is the sum of the seven numbers from 223 to 229.

1582 is the sum of the twenty-eight numbers from 43 to 70.

 

1581 One Teardrop among Millions

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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

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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 × 79.
  • 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√395.
  • 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!

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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

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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

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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.