A Multiplication Based Logic Puzzle

Archive for the ‘Reducible Square Roots’ Category

What’s Special About √1024?

What’s special about √1024? Is it because it and several counting numbers after it have square roots that can be simplified?

Perhaps.

Maybe it is interesting just because √1024 = 32, a whole number. The 5th root of 1024 = 4 and the 10th root of 1024 = 2, both whole numbers as well.

Those equations are true because 32² = 1024, 4⁵ = 1024, and 2¹⁰ = 1024.

Or perhaps 1024 is special because it is the smallest number that is a 10th power. (It is 2¹⁰.) The square root of a perfect 10th power is always a perfect 5th power. (32 = 2⁵ and is the smallest number that is a 5th power.)

1024 is also the smallest number with exactly 11 factors.

It is the smallest number whose factor tree has at least 10 leaves that are prime numbers. (They are the red leaves on the factor tree shown below.) It is possible to draw several other factor trees for 1024, but they will all have the number 2 appearing ten times.

What’s more, I noticed something about 1024 and some other multiples of 256: Where do multiples of 256 fall on the list of square roots that can be simplified?

  • 256 × 1 = 256 and 256 is the 100th number on this list of numbers whose square roots can be simplified.

1st 100 reducible square roots

  • 256 × 2 = 512. When we add the next 100 square roots that can be simplified, 512 is the 199th number on the list.

2nd 100 reducible square roots

  • Here are the third 100 square roots that can be simplified:

Reducible Square Roots 516-765

  • 256 × 3 = 768 didn’t quite make that list because it is the 301st number. Indeed, it is the first number on this list of the fourth 100 numbers whose square roots can be simplified.

  • 256 × 4 = 1024. That will be the first number on the 5th 100 square roots list!

It is interesting that those multiples of 256 have the 100th, the 199th, the 301st, and the 401st positions on the list. That is so close to the 100th, 200th, 300th, and 400th positions.

In case you couldn’t figure it out, the highlighted square roots are three or more consecutive numbers that appear on the list.

1024 is interesting for many other reasons. Here are a few of them:

(4-2)¹⁰ = 1024, making 1024 the 16th Friedman number.

I like to remember that 2¹⁰ = 1024, which is just a little bit more than a thousand. Likewise 2²⁰ = 1,048,576 which is about a million. 2³⁰ is about a billion, and 2⁴⁰ is about a trillion.

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As stated in the comments, Paula Beardell Krieg shared a related post with me. It takes exactly 1024 Legos to build this fabulous pyramidal fractal:

https://platform.twitter.com/widgets.js
1024 has so many factors that are divisible by 4 that it is a leg in NINE Pythagorean triples:
768-1024-1280 which is (3-4-5) times 256
1024-1920-2176 which is (8-15-17) times 128
1024-4032-4160 which is (16-63-65) times 64
1024-8160-8224 which is (32-255-257) times 32
1024-16368-16400 which is 16 times (64-1023-1025)
1024-32760-32776 which is 8 times (128-4095-4097)
1024-65532-65540 which is 4 times (256-16383-16385)
1024-131070-131074 which is 2 times (512-65535-65537),
and primitive 1024-262143-262145

Some of those triples can also be found because 1024 is the difference of two squares four different ways:
257² – 255² = 1024
130² – 126² = 1024
68² – 60² = 1024
40² – 24² = 1024
To find out which difference of two squares go with which triples, add the squares instead of subtracting and you’ll get the hypotenuse of the triple.
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1024 looks interesting in some other bases:
It’s 1000000000 in BASE 2,
100000 in BASE 4,
2000 in BASE 8,
1357 in BASE 9,
484 in BASE 15,
400 in BASE 16,
169 in BASE 29,
144 in BASE 30,
121 in BASE 31, and
100 in BASE 32

  • 1024 is a composite number.
  • Prime factorization: 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, which can be written 1024 = 2¹⁰
  • The exponent in the prime factorization is 10. Adding one we get (10 + 1) = 11. Therefore 1024 has exactly 11 factors.
  • Factors of 1024: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
  • Factor pairs: 1024 = 1 × 1024, 2 × 512, 4 × 256, 8 × 128, 16 × 64, or 32 × 32,
  • 1024 is a perfect square. √1024 = 32. It is also a perfect 5th power, and a perfect 10th power.

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Reducible Square Roots up to √765

  • 765 is a composite number.
  • Prime factorization: 765 = 3 x 3 x 5 x 17, which can be written 765 = (3^2) x 5 x 17
  • 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 x 2 x 2 = 12. Therefore 765 has exactly 12 factors.
  • Factors of 765: 1, 3, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765
  • Factor pairs: 765 = 1 x 765, 3 x 255, 5 x 153, 9 x 85, 15 x 51, or 17 x 45
  • Taking the factor pair with the largest square number factor, we get √765 = (√9)(√85) = 3√85 ≈ 27.658633.

765-factor-pairs

765 is the 300th number whose square root can be reduced! Here are three tables with 100 reducible square roots each showing all the reducible square roots up to √765. When three or more consecutive numbers have reducible square roots, I highlighted them.

1st 100 reducible square roots

2nd 100 reducible square roots

Reducible Square Roots 516-765

That’s 300 reducible square roots found for the first 765 counting numbers. 300 ÷ 765 ≈ 0.392, so 39.2% of the numbers so far have reducible square roots.

Today’s puzzle is a whole lot less complicated than all that, so give it a try!

765 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

Logical steps to find the solution are in a table at the bottom of the post.

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Here are some other fun facts about the number 765:

765 is made from three consecutive numbers so it is divisible by 3. The middle of those numbers is 6 so 765 is also divisible by 9.

765 can be written as the sum of two squares two different ways:

  • 27² + 6² = 765
  • 21² + 18² = 765

Its other two prime factors, 5 and 17, have a remainder of 1 when divided by 4 so 765² can be written as the sum of two squares FOUR different ways, two of which contain other numbers that use the same digits as 765. Also notice that 9 is a factor of each number in the corresponding Pythagorean triples.

  • 117² + 756² = 765²
  • 324² + 693² = 765²
  • 360² + 675² = 765²
  • 459² + 612² = 765²

765 can also be written as the sum of three squares four different ways:

  • 26² + 8² + 5² = 765
  • 22² + 16² + 5² = 765
  • 20² + 19² + 2² = 765
  • 20² + 14² + 13² = 765

765 is a palindrome in two different bases:

  • 1011111101 BASE 2; note that 1(512) + 0(256) + 1(128) + 1(64) + 1(32) + 1(16) + 1(8) + 1(4) + 0(2) + 1(1) = 765.
  • 636 BASE 11; note that 6(121) + 3(11) + 6(1) = 765.

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765 Logic

700 Pick Your Pony! Who will win this Amount of Factors Horse Race?

  • 700 is a composite number.
  • Prime factorization: 700 = 2 x 2 x 5 x 5 x 7, which can be written 700 = (2^2) x (5^2) x 7
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 700 has exactly 18 factors.
  • Factors of 700: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700
  • Factor pairs: 700 = 1 x 700, 2 x 350, 4 x 175, 5 x 140, 7 x 100, 10 x 70, 14 x 50, 20 x 35 or 25 x 28
  • Taking the factor pair with the largest square number factor, we get √700 = (√100)(√7) = 10√7 ≈ 26.457513.

Because this is my 700th post, I think I’ll have another horse race. Some numbers from 601 to 700 have exactly 2 factors, 4 factors, and so forth up to 24 factors. (Only perfect squares can have an odd number of factors.)

Which number from 1 to 24 will win this amount of factors horse race? Which number will come in second place, or third place? Cheering for more than one pony will make the race even more interesting.

Here we see that the numbers 2, 6, & 8 are the first ones out of the gate. Click on the graphic to see the rest of this very thrilling horse race:

605

Every hundred posts I also like to focus on the percentage of numbers whose square roots can be simplified.

700 is divisible by 100 so its square root can easily be simplified: √700 = 10√7.

273 of the first 700 numbers have reducible square roots. That’s exactly 39%.

The rest of the numbers, 427, which is 41% of the first 700 numbers, do not have reducible square roots.

Here’s a table breaking down the amount of factors in each group of one hundred integers and the number of reducible square roots.

1-700 Amount of Factors

Here are some facts about the number 700.

700 is a palindrome in several bases:

  • 4A4 BASE 12; note A is equivalent to 1o in base 10, and 4(144) + 10(12) + 4(1) = 700
  • PP BASE 27; note P is equivalent to 25 in base 10, and 25(27) + 25(1) = 700
  • KK Base 34; note K is equivalent to 20 in base 10, and 20(34) + 20(1) = 700

700 is the sum of four consecutive prime numbers: 167 + 173 + 179 + 181.

Here is a beautiful painting of a horse race that I saw on twitter:

Reducing √588 and Level 3

88 is divisible by 4 so 588 is also divisible by 4, and that means √588 can be reduced.

About 83% of the numbers that have reducible square roots are divisible by 4 and/or by 9, and it is so easy to tell if even a very long number is divisible by either of those numbers. It is also easier to divide a number by 4 or 9 than it is to divide by their square roots twice.

When I reduce a square root, I like to make a little cake and start by dividing by 100, 4, or 9 if any of those numbers are its factors. Here are the steps I used to make a cake for 588 with as many perfect squares on the outside of the cake as possible.

  1. 588 ÷ 4 = 147
  2. 147 is not divisible by 4 again, but 5 + 8 + 8 = 21 so 147 is divisible by 3, but not by 9.
  3. 147 ÷ 3 = 49 which is a perfect square, so I stop dividing and simply take the square roots of everything on the outside of the cake and multiply them together.

This is what my cake looks like:

588 cake

And now for today’s Level 3 puzzle:

588 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 588 is a composite number.
  • Prime factorization: 588 = 2 x 2 x 3 x 7 x 7, which can be written 588 = (2^2) x 3 x (7^2)
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 x 2 x 3 = 18. Therefore 588 has exactly 18 factors.
  • Factors of 588: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588
  • Factor pairs: 588 = 1 x 588, 2 x 294, 3 x 196, 4 x 147, 6 x 98, 7 x 84, 12 x 49, 14 x 42 or 21 x 28
  • Taking the factor pair with the largest square number factor, we get √588 = (√196)(√3) = 14√3 ≈ 24.248711

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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. 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.

588 Factors

Reducible Square Roots Up to √513

5 + 1 + 3 = 9 so 513 can be evenly divided by 9, and thus its square root can be simplified.

513 is the 200th counting number whose square root can be reduced. 200/513 ≈ .38986, which means, so far, 38.97% of the counting numbers have reducible square roots.

Here are the first 100 reducible square roots followed by the second hundred:

1st 100 reducible square roots

I highlighted the ones that are part of three or more consecutive reducible square roots.

2nd 100 reducible square roots

  • 513 is a composite number.
  • Prime factorization: 513 = 3 x 3 x 3 x 19, which can be written 513 = (3^3) x 19
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 513 has exactly 8 factors.
  • Factors of 513: 1, 3, 9, 19, 27, 57, 171, 513
  • Factor pairs: 513 = 1 x 513, 3 x 171, 9 x 57, or 19 x 27
  • Taking the factor pair with the largest square number factor, we get √513 = (√9)(√57) = 3√57 ≈ 22.6495033

How to Simplify √450

What’s easier – dividing a number by 2 twice or dividing by 4 once? Most people would agree that dividing by a single digit number like 9 one time is easier than dividing by 3 two times. It also cuts the chance of making a mistake in half.

To simplify square roots, I’ve modified the cake method to look for some specific SQUARE factors rather than beginning with ALL of its prime factors.

Divisibility tricks let me know which square factors to try. I always divide out easy and very common perfect squares 100, 4, 9, and 25 first. (About 82% of reducible square roots are divisible by 4 and/or 9.) Once any of those that apply have been divided out, I look to see if 6 or 10 can be divided out because its easier to divide by 6 or 10 once than to divide by 2 and then by 3 or 5. Only after I have divided out those very easy divisors will I look to divide out any remaining prime factors 2, 3, 5, 7, 11, and so forth. Let me demonstrate this method to simplify √450.

  • 450 doesn’t end in 00, so it’s not divisible by 100.
  • The last two digits, 50, are not divisible by 4, so 450 is not divisible by 4.
  • 4 + 5 + 0 = 9, so 450 is divisible by 9. Therefore, I do a simple division problem, 450 ÷ 9 = 50, leaving room on the page to do any other needed division problems above it.
  • The previous quotient, 50, is not divisible by 9, but any number ending in 25, 50, or 75 is divisible by 25, so I divide it by 25 and get 2. Now I am finished dividing.
  • The numbers on the outside of the cake are 9, 25, and 2. I take the square root of each of those numbers and get 3, 5, and √2. The product of those square roots is 15√2. Thus √450 = 15√2.

450 Cake

If the number had been something larger like 925, I could multiply 925 by .04 to get 37. Multiplying by .04 is the same as dividing by 25.

Most people have been taught to use a factor tree to find square roots. This is probably 450’s most common factor tree:

450 Factor Tree

I like how much more compact and clear this modified cake method is instead. It may take some practice to get used to it, but I will show more examples of it in the future to make it more familiar.

The following puzzle doesn’t have anything to do with the number 450 except it’s the number I gave it to distinguish it from every other puzzle I make. I put the factors of 450 immediately after the puzzle to separate the puzzle from its solution that I add to the post the next day.

450 Puzzle

Print the puzzles or type the factors on this excel file:  12 Factors 2015-04-06

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  • 450 is a composite number.
  • Prime factorization: 450 = 2 x 3 x 3 x 5 x 5, which can be written 450 = 2 x (3^2) x (5^2)
  • 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 x 3 x 3 = 18. Therefore 450 has exactly 18 factors.
  • Factors of 450: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450
  • Factor pairs: 450 = 1 x 450, 2 x 225, 3 x 150, 5 x 90, 6 x 75, 9 x 50, 10 x 45, 15 x 30 or 18 x 25
  • Taking the factor pair with the largest square number factor, we get √450 = (√225)(√2) = 15√2 ≈ 21.2132

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450 Factors

How to Reduce √432

432 factor tree

Many people use factor trees and prime factorizations to figure out how to reduce square roots. I don’t. I don’t see the point in breaking something completely apart just to put it back together again, especially when it ISN’T necessary to break it completely apart.

So instead of using a factor tree, I’ve modified the cake method, and I use it to reduce square roots.

For example, to find the square root of 500, I would never use its prime factorization: 500 = 2 x 2 x 5 x 5 x 5. Instead I would first divide 500 by perfect square 100 to get 5. Then I would take the square root of both 100 and 5 to get 10√5.

Only 1% of numbers are divisible by 100, but for the ones that are, I always start by dividing by 100.

Now get this: Roughly 82.5% of all numbers that have reducible square roots can be evenly divided by perfect squares four and/or by nine. It is so easy to tell if a number can be evenly divided by either of those numbers. That is why I always start with one hundred, then four, then nine.

Let’s look at 432: Since the number formed from the last two digits, 32, is divisible by 4, I know that 432 is also divisible by four, so I will go ahead and do the division.

432 divided by 4

Now I look at 108. The last two digits, 08, can also be evenly divided by 4, so I do that division as well:

108 divided by 4

108 divided by 4 gives us 27 which is not divisible by 4. However since 2 + 7 = 9, I know that 27 is divisible by 9, so I do that division next:

27 divided by 9

27 divided by 9 is 3. Since the only square number that will divide evenly into 3 is 1, I’m done with the division process. Now I take the square roots of all the numbers on the outside of the cake and multiply them together: √(4 x 4) x (√9) x (√3) = 4 x 3 x (√3) = 12√3.

I will give other examples of this method in future posts. Here’s today’s puzzle:

432 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 432 is a composite number.
  • Prime factorization: 432 = 2 x 2 x 2 x 2 x 3 x 3 x 3, which can be written 432 = (2^4) x (3^3)
  • The exponents in the prime factorization are 4 and 3. Adding one to each and multiplying we get (4 + 1)(3 + 1) = 5 x 4 = 20. Therefore 432 has exactly 20 factors.
  • Factors of 432: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, 432
  • Factor pairs: 432 = 1 x 432, 2 x 216, 3 x 144, 4 x 108, 6 x 72, 8 x 54, 9 x 48, 12 x 36, 16 x 27, or 18 x 24
  • Taking the factor pair with the largest square number factor, we get √432 = (√144)(√3) = 12√3 ≈ 20.7846

432 Logic

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