## A Multiplication Based Logic Puzzle

### 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 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.

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!

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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### 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:

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.

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:

And now for today’s Level 3 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.

### 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:

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

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:

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.

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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### How to Reduce √432

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.

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 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 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:

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

### Reducible Square Roots of Numbers up to 352

• 352 is even and therefore a composite number.
• Prime factorization: 352 = 2 x 2 x 2 x 2 x 2 x 11, which can be written 352 = (2^5) x 11
• The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 352 has exactly 12 factors.
• Factors of 352: 1, 2, 4, 8, 11, 16, 22, 32, 44, 88, 176, 352
• Factor pairs: 352 = 1 x 352, 2 x 176, 4 x 88, 8 x 44, 11 x 32, or 16 x 22
• Taking the factor pair with the largest square number factor, we get √352 = (√16)(√22) = 4√22 ≈ 18.762

Mathematics is full of interesting patterns. Let’s explore some patterns in reducible square roots.

Here are the first ten numbers that have reducible square roots. Notice that five of the numbers in this list, 1, 4, 9, 16, and 25 are perfect squares:

40% of the numbers up to 5 have reducible square roots. The same thing is true for 40% of the numbers up to 10, 40% of the numbers up to 20, and 40% of the numbers up to 25.

Here are the second ten numbers with reducible square roots. The last number,50, is double the previous last number, 25. Again 40% of the numbers up to 50 have reducible square roots, and only two of these numbers are perfect squares. Notice that the last three numbers under the radical sign in this set are consecutive numbers.

Would you like to make any predictions for the third set of ten numbers with reducible square roots?

If you predicted that this set of numbers would end with 75, you were almost right! 29/75 or 38.67% of the numbers up to 75 have reducible square roots, and 30/76 or 39.47% of the numbers up to 76 do. Both of these values are very close to 40%, but not quite there. Notice this time only one number is a perfect square.

Would you like to make a prediction for what will happen with the fourth set of ten numbers with reducible square roots?

Surprise! We’re back to 40% of the numbers up to 100 have reducible square roots. Those consecutive numbers at the end of the set really helped raise the percentage right at the last minute. This set of numbers has two perfect squares.

What do you think will happen if we look at all the reducible square roots up to 350? Multiples of any perfect square will always have reducible square roots. Since we have more perfect squares, do you think more will be reducible?

Because the square roots of 351 and 352 are also reducible, let’s include them in this chart. Each column has 20 reducible square roots in it, and they are grouped into fives for easier counting. I’ve highlighted sets of three or four consecutive numbers. In all, we now have charts showing the first 140 reducible square roots. This last set has eight perfect squares. Let’s look at the percentages at the end of some of those sets of consecutive numbers: 50/126 or 39.68% of the first 126 numbers have reducible square roots. 96/245 or 39.18% of the first 245 numbers have reducible square roots. Finally, 140/352 or 39.77% of the numbers up to 352 have reducible square roots. That one is so close to 40%!

Sets of 41-60, 61-80, 81-100, 101-120, & 121-140 Reducible Square Roots

No matter how big a table we make, the percentage of reducible square roots will be very close to 40%. It will not get significantly higher because, believe it or not, most numbers are either prime numbers or the product of two or more DIFFERENT primes and thus have square roots that are NOT reducible.