factor pairs

How Much of My $1596 Will the Taxman Get?

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Today’s Puzzle is a Game Called Taxman:

Tax season in the United States began just a few days ago. Figuring taxes is no fun, but playing Taxman is.

You see 56 cards below. Imagine that they are 56 envelopes, each containing the dollar amount written on them.

You can take whatever envelope you’d like, but the taxman must be able to take at least one envelope when you do. What envelopes will the taxman take when it is his turn? He will take EVERY available envelope that has a number on it that is a factor of the envelope you took. So if you started off taking envelope 56, the taxman would take envelopes 1, 2, 4, 7, 8, 14, and 28. Thus the taxman would start off getting a whole lot more money than you while also limiting envelopes you can choose later. For example, you wouldn’t be able to pick any prime numbers after taking 56  because the 1 would be gone, and the taxman must be able to take one envelope with every choice you make. When you get to the point that there are no more envelopes that you can take, the taxman gets to take ALL the remaining envelopes. You want the taxman to get less money than you do, and hopefully as little money as possible. I’ve included the factors of each of the numbers at the top of each “envelope” to make the game easier for both you and the taxman. You can print “envelopes” from this excel file Taxman & 1537-1544.  You can have a friend be the taxman as you play or you can play both roles.

How I Played the Game:

I took envelopes as I asked myself these questions:

What is the largest prime number on the table? 53. I take 53, and the only envelope the taxman gets is 1.

What is the largest number that is a prime number squared? 49. I take 49, and the only envelope left for the taxman to take is 7.

What is the largest multiple of 7 that has only one other available factor? 35. I take 35, and the only envelope the taxman can still get is 5.

What is the largest multiple of 5 that has only one other available factor? and so forth.

Here is the order I took the envelopes. It is not the only possible order to use, but it was one in which there were only nine cards at the end of the game for the taxman to claim.

The winner of the game is the one with the most money at the end of the game.

How Do We Know Who Won the Game?

We could add up all the money I got and compare it with all the money the taxman got, but that wouldn’t be much fun. We could add up all the taxman money and subtract it from 1596 to find my total, but I prefer a different way: I remove all the taxman’s money from the table and try to mix envelopes to match mine. I start with my envelope with the most cash, 56 = 8 + 1 + 47, so I put the taxman’s envelopes with those numbers next to my 56.

Matching the “envelopes” up like this will take some time, but it feels like playing a game. It may have taken me as long to play this part of the game as it took me to play Taxman because sometimes I had to remove perfectly good sums in order to use all the taxman’s envelopes in a sum. Here is how I matched the envelopes:

I know that I won the game because I had at least one envelope without a sum next to it. I can determine by  how much I won the game by adding 35 + 33 + 46 + 39 + 27 + 38 + 20 + 30 + 42. That sum tells me I won the game by $310.

How much did the taxman get?
(1596-310)/2 = 643.

How much did I get to keep?
643 + 310 = 953.

A Factor Tree for 1596:

If you know that 40² = 1600, and 1600 – 4 = 1596, we might recognize that 1596 is the difference of two perfect squares: 40² – 2². Then we can also know that
1596 = (40 – 2)(40 + 2) = 38 × 42.

Let’s make a factor tree from the factor pair 38 × 42:

Since 1596 has several factors, it has several other possible factor trees.

Factors of 1596:

  • 1596 is a composite number.
  • Prime factorization: 1596 = 2 × 2 × 3 × 7 × 19, which can be written 1596 = 2² × 3 × 7 × 19.
  • 1596 has at least one exponent greater than 1 in its prime factorization so √1596 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1596 = (√4)(√399) = 2√399
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1596 has exactly 24 factors.
  • The factors of 1596 are outlined with their factor pair partners in the graphic below.

More about the Number 1596:

The reason that I was dividing up $1596 between me and the taxman is that 1596 is the 56th triangular number. It is the 56th triangular number because
(56)(57)/2 = 1596.

1596 is the difference of two squares in four different ways:

1596 is the sum of the first 15 Fibonacci numbers, which also means it is one number less than the 17th Fibonacci number.

 

 

1595 and Level 2

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Today’s Puzzle:

This puzzle is just a multiplication table whose missing factors are not in the usual order. Can you figure out where the factors from 1 to 10 should go?

Factors of 1595:

1595 ends with a 5, so it is divisible by 5.
1 – 5 + 9 – 5 = 0, so 1595 is divisible by 11.

  • 1595 is a composite number.
  • Prime factorization: 1595 = 5 × 11 × 29.
  • 1595 has no exponents greater than 1 in its prime factorization, so √1595 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 1595 has exactly 8 factors.
  • The factors of 1595 are outlined with their factor pair partners in the graphic below.

More about the Number 1595:

1595 is the hypotenuse of FOUR Pythagorean triples:
187-1584-1595, which is 11 times (17-144-145),
264-1573-1595, which is 11 times (24-143-145),
957-1276-1595, which is (3-4-5) times 319, and
1100-1155-1595, which is (20-21-29) times 55.

1594 and Level 1

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Today’s Puzzle:

Write the numbers from 1 to 10 in the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1594:

  • 1594 is a composite number.
  • Prime factorization: 1594 = 2 × 797.
  • 1594 has no exponents greater than 1 in its prime factorization, so √1594 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 1594 has exactly 4 factors.
  • The factors of 1594 are outlined with their factor pair partners in the graphic below.

More about the Number 1594:

1594 is the sum of two squares as well as the double of two squares:
37² + 15² = 1594.
2(26² + 11²) = 1594.

1594 is the hypotenuse of a Pythagorean triple:
1110-1144-1594, calculated from 2(37)(15), 37² – 15², 37² + 15².
It is also 2 times (555-572-797).

Sorting the Factors of 1593

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Curious Sort of Factors:

1593 has eight factors. Three-fourths of those factors can be made using some or all of its digits. The other two factors seem to be talking about each other. 27 says 177 has two sevens, and 177 says that 27 has one seven. I’ve sorted them into two categories:

Today’s Puzzle:

Can 1593 be expressed using its four digits exactly once and (), +, -, ×, ÷?
Almost, but not quite:
1593 = 3³ × 59¹. We are not allowed to use the 3 twice.
1593 = 531√9. We can use digits as exponents, but we are not allowed to use the square root symbol.
1593 is not a Friedman number because 1593 = 1593 is the only way we can express it using (), +, -, ×, ÷ and only its own digits as numbers or exponents.

Now try this: Can you express each of 1593’s eight factors using all four of its digits exactly once and only (), +, -, ×, ÷? I’ve done a few of them to get you started. (One of them can’t be done, and there is more than one possibility for some of them.)

1 = 1⁵⁹³
3 =
9 = 3(9-5-1)
27 =
59 =
177 =
531 =
1593 = 1593. That’s all we can do for that one.

Factors of 1593:

  • 1593 is a composite number.
  • Prime factorization: 1593 = 3 × 3 × 3 × 59, which can be written 1593 = 3³ × 59.
  • 1593 has at least one exponent greater than 1 in its prime factorization so √1593 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1593 = (√9)(√177) = 3√177.
  • 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 1593 has exactly 8 factors.
  • The factors of 1593 are outlined with their factor pairs in the graphic below.

More about the Number 1593:

Consecutive numbers 1592 and 1593 each have eight factors because they are each a cube times a prime.

1593 is the difference of two squares four different ways:
797² – 796² = 1593,
267² – 264² = 1593,
93² – 84² = 1593, and
43² – 16² = 1593.

1592 One More Valentine

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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.

1591 Conversation Heart

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Today’s Puzzle:

If this valentine-shaped level 6 puzzle gets kids talking about multiplication, then it will truly be a conversation heart.

Factors of 1591:

  • 1591 is a composite number.
  • Prime factorization: 1591 = 37 × 43.
  • 1591 has no exponents greater than 1 in its prime factorization, so √1591 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 1591 has exactly 4 factors.
  • The factors of 1591 are outlined with their factor pair partners in the graphic below.

More about the Number 1591:

1591 is the difference of two squares two different ways:
796² – 795² = 1591, and
40² – 3² = 1591. That means we are only 3², or 9 numbers away from the next perfect square, 40², or 1600.

1590 A Single Rose

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Today’s Puzzle:

A single rose can be an elegant expression of affection. This single rose is a level 5 puzzle. Can you find its factors?

Here’s the same puzzle without any added color:

Two Factor Trees for 1590:

There are several possible factor trees for 1590. Here are two of them.

Factors of 1590:

  • 1590 is a composite number.
  • Prime factorization: 1590 = 2 × 3 × 5 × 53.
  • 1590 has no exponents greater than 1 in its prime factorization, so √1590 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 1590 has exactly 16 factors.
  • The factors of 1590 are outlined with their factor pair partners in the graphic below.

More about the Number 1590:

1590 is the hypotenuse of FOUR Pythagorean triples:
138-1584-1590, which is 6 times (23-264-265),
576-1482-1590, which is 6 times (96-247-265)
840-1350-1590, which is 30 times (28-45-53),
954-1272-1590, which is (3-4-5) times 318.

 

1589 Candy Bars

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Today’s Puzzle:

Perhaps you can imagine that this level 4 puzzle looks like a couple of candy bars, one for you and one for me!

Factors of 1589:

Divisibility rules let us know quickly that 1589 is not divisible by 2, 3, or 5. Is it divisible by 7, the next prime number. We can apply the divisibility rule for 7 to find out:

1589 is divisible by 7 because
158 – 2(9) = 158 – 18 = 140, and 140 is divisible by 7.

  • 1589 is a composite number.
  • Prime factorization: 1589 = 7 × 227.
  • 1589 has no exponents greater than 1 in its prime factorization, so √1589 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 1589 has exactly 4 factors.
  • The factors of 1589 are outlined with their factor pair partners in the graphic below.

More about the Number 1589:

1589 is the difference of two squares two different ways:
795² – 794² = 1589, and
117² – 110² = 1589.

1589 is the sum of consecutive numbers in three different ways:
1589 is the sum of the two numbers from 794 to 795.
1589 is the sum of the seven numbers from 224 to 230.
1589 is also the sum of the fourteen numbers from 107 to 120,

 

1588 Cupid’s Arrow

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Today’s Puzzle:

Keeping with our Valentine’s theme, today’s level 3 puzzle looks like Cupid’s Arrow. Start with the clues at the top of the arrow, write in their factors, and work your way down the puzzle, cell by cell, writing in factors as you go. Before long, you will be smitten with this puzzle!

Factors of 1588:

  • 1588 is a composite number.
  • Prime factorization: 1588 = 2 × 2 × 397, which can be written 1588 = 2² × 397.
  • 1588 has at least one exponent greater than 1 in its prime factorization so √1588 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1588 = (√4)(√397) = 2√397.
  • 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 1588 has exactly 6 factors.
  • The factors of 1588 are outlined with their factor pair partners in the graphic below.

More about the number 1588:

1588 is the difference of two squares:
398² – 396² = 1588.

1588 is also the sum of two squares:
38² + 12² = 1588.

1588 is the hypotenuse of a Pythagorean triple:
912-1300-1588, calculated from 2(38)(12), 38² – 12², 38² + 12².
It is also 4 times (228-325-397).

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.