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 ×
  • 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!

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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.

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

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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.

Sharing $15.75 Worth of Puzzles

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Pattern Puzzle:

When I saw that 35 × 45 = 1575, I suspected a pattern. I made a chart to see if my suspicions were true, and they were! Can you look at the chart and tell me what that pattern is?

If you were able to see that pattern, then look at each of these. They have patterns because the numbers in 3 × 17, 4 × 16, 5 × 15, 6 × 14, and 7 × 13 have a relationship. What is that relationship?

Factors of 1575:

We can use 35 × 45 = 1575 to make one of its many possible factor trees:

  • 1575 is a composite number.
  • Prime factorization: 1575 = 3 × 3 × 5 × 5 × 7, which can be written 1575 = 3² × 5² × 7.
  • 1575 has at least one exponent greater than 1 in its prime factorization so √1575 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1575 = (√225)(√7) = 15√7.
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1575 has exactly 18 factors.
  • The factors of 1575 are outlined with their factor pair partners in the graphic below.

Can you use a different factor pair to create another factor tree for 1575? Will you always get 1575 = 3² × 5² × 7 in the end?

Difference of Two Squares Puzzle:

1575 is the difference of two squares in NINE different ways:
788² – 787² = 1575,
264² – 261² = 1575,
160² – 155² = 1575,
116² – 109² = 1575,
92² – 83² = 1575,
60² – 45² = 1575,
48² – 27² = 1575,
44² – 19² = 1575, and
40² – 5² = 1575.

In money 1575¢ is represented as $15.75. That’s the same as 63 quarters! Which of those differences of two squares is illustrated using quarters in the image below:

Which of the nine difference of two squares above is illustrated in the following image?

That image illustrates that $15.75 is just one quarter away from the next perfect square dollar amount, $16.00. Both 16 and 1600 are perfect squares. Can you make the rectangle below by moving just one row of quarters from the image above?

Moving that one row could help you notice that
8² – 1² = (8 – 1) × (8 + 1) = 63,
and might be the first step in understanding that  a² – b² = (a + b)(a – b) .

Dividing Mixed Numbers Puzzles:

A quarter is 25¢. The reason a quarter is called a quarter is because it is a quarter or 1/4th of a dollar. We usually write dollar and cents together as decimals. A quarter is $0.25.

Three quarters is 75¢ or $0.75 and is 3/4ths of a dollar.

Two quarters is 50¢ or $0.50 and is 2/4ths or one half of a dollar.

Representing 1575¢ in quarters can help you understand dividing mixed numbers like in the problem below:

The answer to both questions is the same! Now try this one:

You might not find this next example easy, but give it a look:

Why is 13 in the denominator of the answer to both questions when it didn’t appear in either question? Where did the 13 come from?

Now try writing and solving your own problem:

Working with money often seems like more fun than working with numbers. I hope you enjoyed these puzzles today.

 

1574 Avoiding Fractions When Using the Substitution Method to Solve Simultaneous Equations

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The Substitution Method

Jo Morgan examines six methods of solving simultaneous linear equations in her wonderful book A Compendium of Mathematical Methods. Six is two more than the number of methods I had known previously. Jo loves pouring over vintage mathematics books to learn how concepts were taught in days gone by. She loves finding resources on Twitter to teach concepts and describes herself as a Resourceaholic. Today is her birthday, and this post is dedicated to her.

In my experience, the substitution method is often the first method taught to solve simultaneous linear equations, and usually, before the accompanying homework assignment is finished, fractions will be part of the solution process. Once students learn other methods, they usually abandon the substitution method with its troublesome fractions. However, the substitution method is really just as good as any other method, and fractions can actually be avoided when using it! I encourage students to use the substitution method and avoid fractions while they work.

In the table below I use the substitution method twelve ways to find one coordinate of the solution. (In each case the other coordinate could be found by substituting the known value back into one of the original equations and solving for the other coordinate.) Notice that only the first column uses fractions; the other two columns do not.

Follow each solution process step by step. Do any of the methods seem less confusing than the others? How will you approach the problem the next time you are asked to solve simultaneous linear equations?

Factors of 1574:

This is my 1574th post. Here is some information about that number:

  • 1574 is a composite number.
  • Prime factorization: 1574 = 2 × 787.
  • 1574 has no exponents greater than 1 in its prime factorization, so √1574 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 1574 has exactly 4 factors.
  • The factors of 1574 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1574:

1574 is in just one Pythagorean triple:
1574-619368-619370, calculated from 2(787)(1), 787²-1², 787²+1².
It can also be calculated from 2(394²-393²), 4(394)(393), 2(394²+393²).

1573 Ring Out Wild Bells

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Today’s Puzzle:

Here are the first two verses of Alfred Lord Tennyson’s Ring Out Wild Bells:

Ring out, wild bells, to the wild sky,
The flying cloud, the frosty light;
The year is dying in the night;
Ring out, wild bells, and let him die.

Ring out the old, ring in the new,
Ring, happy bells, across the snow:
The year is going, let him go;
Ring out the false, ring in the true.

I think we are all ready for 2020 to “die” so we can ring in 2021. You can get started with this bell-shaped puzzle. Use logic to find its unique solution.

Print the puzzles or type the solution in this excel file: 12 Factors 1558-1573

Factors of 1573:

This is my 1573rd post, so let’s find the factors of 1573.

  • 1573 is a composite number.
  • Prime factorization: 1573 = 11 × 11 × 13, which can be written 1573 = 11² × 13
  • 1573 has at least one exponent greater than 1 in its prime factorization so √1573 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1573 = (√121)(√13) = 11√13
  • 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 1573 has exactly 6 factors.
  • The factors of 1573 are outlined with their factor pair partners in the graphic below.

More about the number 1573:

1573 is the sum of two squares:
33² + 22² = 1573.

1573 is the hypotenuse of a Pythagorean triple:
605-1452-1573, calculated from 33² – 22², 2(33)(22), 33² + 22².
It is also (5-12-13) times 121.