A Multiplication Based Logic Puzzle

Archive for the ‘Mathematics’ Category

882 Factor Trees for the First Day of Autumn

September 22 was the first day of autumn. Leaves are already beginning to fall from the trees.

To rake up the leaves for 882, you might first notice that it’s even. The logical thing to do would be to first divide 882 by two. . . But perhaps you might notice that 8 + 8 + 2 = 18, a number divisible by nine, so you might just as logically begin by dividing 882 by 9. The first step you take determines how the factor tree looks.

882 has many possible factor trees but these two are probably the most common.

You can rake the leaves up this way or you can rake them up that way, but when you rake up the leaves from 882’s factor trees, you always get the same prime factors: 2, 3, 3, 7, and 7.

Here’s a little more about the number 882:

882 has eighteen factors. The greatest number less than 882 with eighteen factors is 828. Now get this: 288 also has eighteen factors. That means that every possible combination of 8-8-2 has exactly eighteen factors!

882 has some interesting representations in some other bases:

  • 616 BASE 12, because 6(12²) + 1(12)¹ + 6(12º) = 882
  • 242 BASE 20, because 2(20²) + 4(20)¹ + 2(20º) = 882
  • 200 BASE 21, because 2(21²) = 882

882 is also the sum of consecutive primes: 439 + 443 = 882

  • 882 is a composite number.
  • Prime factorization: 882 = 2 × 3 × 3 × 7 × 7, which can be written 882 = 2 × 3² × 7²
  • The exponents in the prime factorization are 1, 2 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1)(2 + 1) = 2 × 3 × 3 = 18. Therefore 882 has exactly 18 factors.
  • Factors of 882: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 441, 882
  • Factor pairs: 882 = 1 × 882, 2 × 441, 3 × 294, 6 × 147, 7 × 126, 9 × 98, 14 × 63, 18 × 49 or 21 × 42
  • Taking the factor pair with the largest square number factor, we get √882 = (√441)(√2) = 21√2 ≈ 29.6984848.

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880 Flip the Diagonals!

There are 880 different 4 × 4 Magic Squares. (In addition, each of those can be rotated and/or reflected to make 880 × 8 Magic Squares.) I have found a few that start with some simple ways to order the numbers 1 to 16 in the boxes and then flip the diagonals to create a perfect Magic Square.

For this first one I began by putting the numbers 1-4 in one box with the numbers 13-16 in the cater-corner box. Then 5-8 and 9-12 occupy the other spaces. Then I flipped the diagonals and got a Magic Square!

If we flip the locations of the numbers 5-8 and the numbers 9-12, it still works:

This third Magic Square begins with the most common way to order the numbers from 1 to 16:

With just a little tweaking of that most common way to order the numbers, I was able to make the following three Magic Squares. Begin by switching the first two rows with each other and the last two rows with each other. Notice in every beginning square the diagonals already equal 34.

We can also switch the first two rows with the last two rows, but that just gives us a reflection of the one just above it. (The colors don’t match, but it is still a reflection!)

Try switching the first two columns with the last two columns. You’ll get another different one.

There are MANY more ways to do a 4 × 4 Magic Square.

Last week’s excel file, 12 factors 864-874, included some Magic Square templates including the 4 × 4 one. You can use that file to find more Magic Squares. Go ahead give it a try!

Here’s a little more about the number 880:

880 is the hypotenuse of a Pythagorean triple:

  • 528-704-880, which is (3-4-5) times 176.

880 is the sum of the twelve prime numbers from 47 to 101.

  • 880 is a composite number.
  • Prime factorization: 880 = 2 × 2 × 2 × 2 × 5 × 11, which can be written 880 = 2⁴ × 5 × 11
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 880 has exactly 20 factors.
  • Factors of 880: 1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 40, 44, 55, 80, 88, 110, 176, 220, 440, 880
  • Factor pairs: 880 = 1 × 880, 2 × 440, 4 × 220, 5 × 176, 8 × 110, 10 × 88, 11 × 80, 16 × 55, 20 × 44 or 22 × 40
  • Taking the factor pair with the largest square number factor, we get √880 = (√16)(√55) = 4√55 ≈ 29.66479.

875 Multiplication and Division Facts

This simple illustration of $8.75 can be used to show a variety of multiplication facts. We can easily see that this much money can be divided by 5, 7, 25, and even 125.

Likewise, if we know that the amount of money is $8.75, we can write that as 8¾ and do some other easy division problems if we divide the money into groups:

  • 8¾ ÷ 1¼ = 7 because 1¼ is 5 quarters.
  • 8¾ ÷ 1¾ = 5 because 1¾ is 7 quarters.

Now, it might not be obvious that there really is $8.75 in money unless we divide the money into groups of four quarters.

Since $8.75 is the same as 8¾, we can try this division problem, too:

How much is 8¾ ÷ 1½?

What other fractional division problems can you do with these 35 quarters?

Here’s a little more about the number 875:

875 is the hypotenuse of three Pythagorean triples:

  • 525-700-875 which is (3-4-5) times 175.
  • 245-840-875 which is (7-24-25) times 35.
  • 308-819-875 which is 7 times (44-117-125)

Factors of 875 were shown in red in those triples. Here are all of 875’s factors:

  • 875 is a composite number.
  • Prime factorization: 875 = 5 × 5 × 5 × 7, which can be written 875 = 5³ × 7
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 875 has exactly 8 factors.
  • Factors of 875: 1, 5, 7, 25, 35, 125, 175, 875
  • Factor pairs: 875 = 1 × 875, 5 × 175, 7 × 125, or 25 × 35
  • Taking the factor pair with the largest square number factor, we get √875 = (√25)(√35) = 5√35 ≈ 29.5803989

871 Is the Smallest Number That Is the Sum of 3 Triangular Numbers 23 Different Ways

Stetson.edu alerted me to the fact that 871 is the smallest number that is the sum of three triangular numbers 23 different ways. I set out to find those 23 ways. We’ll have to count 0 as the 0th triangular number for four of them, but here are those record-breaking 23 ways:

871 is the hypotenuse of Pythagorean triple 335-804-871 which is (5-12-13) times 67.

871 is also repdigit 111 in BASE 29 because 1(29²) + 1(29¹) + 1(29º) = 871.

  • 871 is a composite number.
  • Prime factorization: 871 = 13 × 67
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 871 has exactly 4 factors.
  • Factors of 871: 1, 13, 67, 871
  • Factor pairs: 871 = 1 × 871 or 13 × 67
  • 871 has no square factors that allow its square root to be simplified. √871 ≈ 29.512709

870 If You Can Solve a 3×3 and a 4×4 Magic Square, Then You Can Solve a 12×12 Magic Square

It’s simple multiplication, folks!

3×4 = 12, so if you can solve a 3×3 Magic Square and a 4×4 Magic Square, then you can solve a 12×12 Magic Square, too.

Seriously. Really. It’s true! You really can solve a 12×12 Magic Square. I did it. Twice. You’ll see. You can do it, too.

That solution consists of sixteen 3×3 Magic Squares that use all the numbers from 1 to 144. The one in the corner is the most familiar one. The other ones are just that familiar Magic Square plus 9, 18, 27 and a few other multiples of 9. I used the 4×4 Magic Square below as a guide as I placed the sixteen 3×3 Magic Squares on the excel file I made, 12 factors 864-874.

4 x 4 Magic Square

make science GIFs like this at MakeaGif

There is 1 solution times 8 orientations (right-side-up, upside-down, etc.) for the 3×3 Magic Square, and there are 880 solutions times 8 orientations for the 4×4 Magic Square. That’s a lot of different possible solutions!

I also solved the 12×12 Magic Square by using nine 4×4 Magic Squares and placing them on the second grid in 12 factors 864-874. This arrangement also has many more possibilities than the one shown here.

You can easily solve your own 12×12 Magic Square by opening this excel file, 12 factors 864-874, going to the 12×12 Magic Square tab, and following the directions.

Here’s more about the awesome number 870:

29 × 30 = 870. That means 870 is 29 × (29+1), so 870 is the sum of the first 29 even numbers.

870 is the sum of the ten prime numbers from 67 to 107.

Because 5 and 29 are its factors, 870 is the hypotenuse of four Pythagorean triples: ( Note that 5 × 29 = 145)

  • 102-864-870, which is 6 times (17-144-145)
  • 144-858-870, which is 6 times (24-143-145)
  • 522-696-870, which is (3-4-5) times 174
  • 600-630-870, which is (20-21-29) times 30

870 has some interesting representations in other bases:

  • 606 in BASE 12, because 6(12²) + 0(12¹) + 6(12º) = 6(145) = 870
  • 303 in BASE 17, because 3(17²) + 0(17¹) + 3(17º) = 3(290) = 870
  • 110 in BASE 29, because 1(29²) + 1(29¹) + 0(29º) = 870
  • T0 in BASE 30 (T is 29 base 10), because 29(30¹) + 0(30º) = 29(30) = 870

8 + 0 + 7 + 8³ + 0³ + 7³ = 870. Thank you Stetson.edu for that fun fact.

  • 870 is a composite number.
  • Prime factorization: 870 = 2 × 3 × 5 × 29
  • 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 × 2 × 2 × 2 = 16. Therefore 870 has exactly 16 factors.
  • Factors of 870: 1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870
  • Factor pairs: 870 = 1 × 870, 2 × 435, 3 × 290, 5 × 174, 6 × 145, 10 × 87, 15 × 58, or 29 × 30
  • 870 has no square factors that allow its square root to be simplified. √870 ≈ 29.495762

Palindrome 868 Has a Fun Square Root

Obviously, 868 is a palindrome in base 10. It has some interesting representations in some other bases, too:

  • 868 is 4004 in BASE 6, because 4(6³) + 0(6²) + 0(6¹) + 4(6º) = 868
  • SS in BASE 30 (S is 28 base 10), because 28(30) + 28(1) = 28(31) = 868
  • S0 in BASE 31, because 28(31) = 868

From Stetson.edu I learned that all the digits from 1 to 9 make up the first nine decimal places of √868, so I decided to make a gif showing that fun fact:

Square root of 868

make science GIFs like this at MakeaGif
  • 868 is a composite number.
  • Prime factorization: 868 = 2 × 2 × 7 × 31, which can be written 868 = 2² × 7 × 31
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 868 has exactly 12 factors.
  • Factors of 868: 1, 2, 4, 7, 14, 28, 31, 62, 124, 217, 434, 868
  • Factor pairs: 868 = 1 × 868, 2 × 434, 4 × 217, 7 × 124, 14 × 62, or 28 × 31,
  • Taking the factor pair with the largest square number factor, we get √868 = (√4)(√217) = 2√217 ≈ 29.461839725

866 Please Help with the September 2017 Math Education Blog Carnival!

Carnivals are part of most state fairs, including the Utah State Fair.

Here is a mathematically nonsensical commercial encouraging us to “Go Beyond Ordinary at the Utah State Fair – Sept 7 – 17, 2017.” (You will have to click this picture and the facebook post to see the video.)

I will be hosting the September 2017 Math Education Blog Carnival in a couple of weeks! I am excited, but also a bit terrified. I have never done anything like this before, and, fellow bloggers, I would really appreciate your support.

As always, the Math Education Blog Carnival will include posts about math that will make sense and be great fun for teachers, their students, and even parents. If you have a math education blog post, please submit it to this month’s blog carnival! Click here for instructions on how to submit your post. You can also contact me on twitter: Iva Sallay @findthefactors.

Now here’s a little bit about the number 866:

29² + 5² = 866, so 866 is the hypotenuse of a Pythagorean triple:

290-816-866 which is 2(29)(5) , 29² – 5² , 29² + 5² and 2 times (145-408-433)

  • 866 is a composite number.
  • Prime factorization: 866 = 2 × 433
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 866 has exactly 4 factors.
  • Factors of 866: 1, 2, 433, 866
  • Factor pairs: 866 = 1 × 866 or 2 × 433
  • 866 has no square factors that allow its square root to be simplified. √866 ≈ 29.4278779

 

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