A Multiplication Based Logic Puzzle

Posts tagged ‘factor tree’

756 and Level 3

  • 756 is a composite number.
  • Prime factorization: 756 = 2 x 2 x 3 x 3 x 3 x 7, which can be written 756 = (2^2) x (3^3) x 7
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 756 has exactly 24 factors.
  • Factors of 756: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756
  • Factor pairs: 756 = 1 x 756, 2 x 378, 3 x 252, 4 x 189, 6 x 126, 7 x 108, 9 x 84, 12 x 63, 14 x 54, 18 x 42, 21 x 36 or 27 x 28
  • Taking the factor pair with the largest square number factor, we get √756 = (√21)(√36) = 6√21 ≈ 27.495454.

756-factor-pairs

 

756 has many factors and, therefore, it has many possible factor trees. Here are three of them:

756 factor trees

Here’s a level 3 Find the Factors puzzle for you to solve, too:

756 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s a few more thoughts on the number 756:

The last two digits of 756 is divisible by 4 so 756 is divisible by 4.

756 is formed from 3 consecutive numbers (5, 6, 7) so it is divisible by 3. The middle number is divisible by 3 so 756 is also divisible by 9.

756 can be written as the sum of consecutive numbers seven ways:

  • 251 + 252 + 253 = 756; that’s 3 consecutive numbers.
  • 105 + 106 + 107 + 108 + 109 + 110 + 111 = 756; that’s 7 consecutive numbers.
  • 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 = 756; that’s 8 consecutive numbers.
  • 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 756; that’s 9 consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 756; that’s 21 consecutive numbers.
  • 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 = 756; that’s 24 consecutive numbers.
  • 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41  = 756; that’s 27 consecutive numbers.

756 is also the sum of six consecutive prime numbers: 109 + 113 + 127+ 131 + 137+ 139 = 756.

756 can be written as the sum of three squares four different ways. (Notice that all of the squares are even):

  • 26² + 8² + 4² = 756
  • 24² + 12² + 6² = 756
  • 22² + 16² + 4² = 756
  • 20² + 16² + 10² = 756

756 is a palindrome in two other bases:

  • 11011 BASE 5; note that 1(625) + 1(125) + 0(25) + 1(5) + 1(1) = 756.
  • LL BASE 35 (L is 21 base 10); note that 21(35) + 21(1) = 756.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

756 Factors

720 Christmas Factor Trees

720 has more factors than any previous number. It has 30 factors.

6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 720, and is usually written 6! = 720.

Since I hosted a Christmas Eve dinner for my extended family, I thought of some of the many ways people could fill up their plates.

720 Dinners

If we made a tree diagram of all the possible dinners that could be made with exactly one item from each column, it would contain 720 lines and require quite a few pages.

The fundamental counting principle tells us the easiest way to count all those dinners is to multiply together the number of items in each column. In this case that would be 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 720.

On the other hand factor trees for 720 are easy to make.

Here are some of MANY possible factor trees for 720:

720 Some Factor Trees

None of those were very attractive, but here are some that are much better looking:

720 More Factor Trees

Here are some other random facts about the number 720:

Because 5 is one of its factors, 720 is the hypotenuse of the Pythagorean triple 432-576-720. What is the greatest common factor of those 3 numbers? The greatest common factor is in the factor pair with the number 5. All 15 factor pairs for 720 are listed at the end of the post.

720 Hypotenuse

The sum of the interior angles of any hexagon is 720 degrees.

720 degrees hexagons

720 is a palindrome in three different bases:

  • 5A5 in base 11; note that 5(121) + 10(11) + 5(1) = 720.
  • OO in base 29; (O is 24 base 10) note that 24(29) + 24(1) = 720.
  • KK in base 35; (K is 20 base 10) note that 20(35) + 20(1) = 720.

 

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  • 720 is a composite number.
  • Prime factorization: 720 = 2 x 2 x 2 x 2 x 3 x 3 x 5, which can be written 720 = (2^4) x (3^2) x 5
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each and multiplying we get (4 + 1)(2 + 1)(1 + 1) = 5 x 3 x 2 = 30. Therefore 720 has exactly 30 factors.
  • Factors of 720: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720
  • Factor pairs: 720 = 1 x 720, 2 x 360, 3 x 240, 4 x 180, 5 x 144, 6 x 120, 8 x 90, 9 x 80, 10 x 72, 12 x 60, 15 x 48, 16 x 45, 18 x 40, 20 x 36 or 24 x 30

Taking the factor pair with the largest square number factor, we get √720 = (√144)(√5) = 12√5 ≈ 26.8328157.

 

688 is a Friedman number

688 is a palindrome in two different bases:

  • 2002 in base 7; note that 2(343) + 0(49) + 0(7) + 2(1) = 688
  • 494 in base 12; note that 4(144) + 9(12) + 4(1) = 688

688 is called a Friedman number because it can be expressed using only its own digits and +, -, x, ÷, ( ), and exponents. Stetson.edu has published a table of all such numbers that have 4 digits or less and the reason each qualifies to be a Friedman number.

688 is a Friedman number because 688 = 8 x 86 so I made a factor tree based on that single multiplication fact:

688 Factor Tree

Since it’s such a fun number fact, I positioned it on top of today’s factoring puzzle, too.

688 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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  • 688 is a composite number.
  • Prime factorization: 688 = 2 x 2 x 2 x 2 x 43, which can be written 688 = (2^4) x 43
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 688 has exactly 10 factors.
  • Factors of 688: 1, 2, 4, 8, 16, 43, 86, 172, 344, 688
  • Factor pairs: 688 = 1 x 688, 2 x 344, 4 x 172, 8 x 86, or 16 x 43
  • Taking the factor pair with the largest square number factor, we get √688 = (√16)(√43) = 4√43 ≈ 26.229754.

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688 Factors

330 Christmas Factor Trees

  • 330 is a composite number.
  • Prime factorization: 330 = 2 x 3 x 5 x 11
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 330 has exactly 16 factors.
  • Factors of 330: 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330
  • Factor pairs: 330 = 1 x 330, 2 x 165, 3 x 110, 5 x 66, 6 x 55, 10 x 33, 11 x 30, or 15 x 22
  • 330 has no square factors that allow its square root to be simplified. √330 ≈ 18.166

Within these seven factor trees for 330 there are also factor trees for 6, 10, 15, 22, 30, 33, 55, 66, 110, and 165, the tops of which are all in brown. The prime factors of 330 are all in red.

330 Factor Trees

 

Can you find the factors and complete this Christmas tree multiplication table?2014-51 Level 2

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

2014-51 Level 2 Factors

 

 

 

200 and Level 1

200 is a composite number.
Factor pairs: 200 = 1 x 200, 2 x 100, 4 x 50, 5 x 40, 8 x 25, or 10 x 20
Factors of 200: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200
Prime factorization: 200 = 2 x 2 x 2 x 5 x 5, which can be written 200 = (2^3) x (5^2)

200 is the smallest positive number that cannot be turned into a prime number by changing ONE of its digits to a different digit. If you can remember that little fact, if should help you to remember that ALL the numbers from 200 to 210 are composite numbers.

2014-32 Level 1

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-11

2014-32 Level 1 Factors

198 and Level 6

198 is a composite number.
Factor pairs: 198 = 1 x 198, 2 x 99, 3 x 66, 6 x 33, 9 x 22, or 11 x 18
Factors of 198: 1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198
Prime factorization: 198 = 2 x 3 x 3 x 11, which can be written 198 = 2 x (3^2) x 11

Since 1 + 9 + 8 = 18, a multiple of 9, 198 is divisible by 9. Here is one of several possible factor trees for 198:

198 factor tree

2014-31 Level 6

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-04

The common factors of 20 and 40 are 4, 5, and 10. Since 14 and either 8, 10, or 6 will use both 2’s, 10 cannot be the CF. If we choose 4 as the common factor, then 5 and 10 would be placed in the top row followed by 3, the factor of 15. However, then we wouldn’t be able to use either 3 or 6 as the common factor of 24 and 18 because 3 would already be in the top row and 4 would already be in the 1st column.  Therefore, the common factor of 20 and 40 has to be 5.

2014-31 Level 6 Logic

15 is the Magic Sum of a 3 x 3 Magic Square

15 is a composite number. 15 = 1 x 15 or 3 x 5. Factors of 15: 1, 3, 5, 15. Prime factorization: 15 = 3 x 5.

When 15 is a clue in the FIND THE FACTORS 1 – 10 or 1 – 12 puzzles, use 3 and 5 as the factors.

If you added the first nine counting numbers together, what sum would you get? What is 1 + 2 +3 + 4+ 5 + 6 + 7 + 8 + 9?

Would you get the same answer by adding (1 + 9) + (2 + 8) + (3 +7) + (4 + 6) + 5?

These are two of the many fun questions you can explore when you try to make a magic square. What is a magic square? If you can place the numbers from 1 to 9 in the box below so that the sum of any row, column, or diagonal will equal the sum of any other row, column, or diagonal, then you will have made a 3 x 3 magic square. The sum of a row, column, or diagonal in a magic square is called the magic sum.

1-9

Clearly it is not a magic square yet. In fact, only one of the numbers is positioned where it needs to be. Which number do you think is already in the correct position?

When it becomes a magic square, what will the magic sum be? One student noticed that in its current state the sums of the rows are 6, 15, and 24. The sums of the columns are 12, 15, 18. The sums of the diagonals are 15 and 15. Since 15 occurs most often, could the magic sum be 15? One way to determine what the magic sum should be is to add the sums of all three rows and then divide by the number of rows. Since 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 and 45 ÷ 3 = 15, then 15 is indeed the magic sum.

Here are a few easy-to-remember steps to construct a 3 x 3 magic square quickly.

Step 1: Draw a tic-tac-toe board and put 5 in the middle.

step 1 magic

Step 2: Put one of the even numbers in one of the corners.  You have four different choices, 2, 4, 6, or 8. The illustration is for the number 2, but any of the even numbers will work.

step 2 magic

Step 3: Subtract your even number from 10 to find its partner. 4 + 6 are partners and so are 2 + 8. Put the partner of the number you chose for step 1 in the corner that is diagonal to it.

step 1 magic

Step 4: Put the other two even numbers in the remaining corners. Yes, you have two choices where to put the numbers. Either choice will work.

step 4 magic

Step 5: Since 6 + 8 = 14 and 15 – 14 = 1, put 1 in the cell between the 6 and the 8. Do similar addition and subtraction problems on each side of the square to determine where to place the 3, 7, and 9. You can work clockwise or counter clockwise, or skip around the square doing the addition and subtraction problems; it doesn’t matter.

This finished magic square looks like this:

step 5 magic

Check it out! Every row, column, and diagonal adds up to 15!

As we created the square, we made choices. First we chose between 4 even numbers, and later we had 2 more choices. Notice that 4 x 2 = 8. There are 8 different ways to make a 3 x 3 magic square! (However, they are all really the same square turned upside down, rolled on its side, viewed from the back. etc.)

There are 880 different ways to make a 4 x 4 magic square. Look over the related articles at the end of this post to learn more about magic squares that are bigger than 3 x 3.

Speaking of magic squares, when I look at the square logic puzzle below, something magical happens. This puzzle has nine clues in it, and all of them are perfect squares. I can use those nine clues to construct a complete multiplication table. If you finish the same puzzle, your multiplication table will look exactly like mine because this puzzle has only one solution.

2014-01-06.1

The level 3 puzzle below is only a little bit more difficult. To solve it place the numbers 1 – 10 in the top row and again in the first column so that those placed numbers are the factors of the given clues. Again there is only one solution, and you will need to use logic to find it. Click 10 Factors 2014-01-06 for more puzzles and last week’s answers.

2014-01-06.3

May we all find a little bit more magic in our lives!

 

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