Mathematics

383 is in only one Pythagorean triple, and it’s primitive

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A Primitive Threesome https://www.mycomicshop.com/search?TID=243401

Although it would take awhile to count to 383, it is still a relatively small number. If you scroll down, you will see that it is also a prime number. Since all odd numbers greater than one are the short legs of a primitive Pythagorean triple, 383 is also. I will explain how to find that primitive triple using the Flintstones, the primitive yet modern stone age family as an analogy. You can use this method on ANY odd number greater than one to find a primitive triple:

  • 383 is like Pebbles the shortest one in the family
  • If we round down (383^2)/2 or 73344.5 to 73,344, we get a much bigger number, much like Fred is much taller than Pebbles in the family
  • If we round the same expression up, we get 73,345, much like Wilma is just a little taller than Fred.
  • 383 – 73344 – 73345 is a primitive Pythagorean triple just as Pebbles, Fred, and Wilma are a primitive yet modern stone age family of three. Test it out: (383^2) + (73344^2) = (73345^2)

Prime numbers can be one of the legs of only one Pythagorean triple, and it will always be a primitive. Sometimes prime numbers are the hypotenuse of a primitive Pythagorean triple, but 383 isn’t. In a future post, I’ll explain how to determine if a prime number is ever the hypotenuse.

Here is 383’s factoring information:

  • 383 is a prime number.
  • Prime factorization: 383 is prime.
  • The exponent of prime number 383 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 383 has exactly 2 factors.
  • Factors of 383: 1, 383
  • Factor pairs: 383 = 1 x 383
  • 383 has no square factors that allow its square root to be simplified. √383 ≈ 19.570

How do we know that 383 is a prime number? If 383 were not a prime number, then it would be divisible by at least one prime number less than or equal to √383 ≈ 19.570. Since 383 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 383 is a prime number.

378 Do You See a Pattern?

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Most people know that 378 is a composite number because it’s obviously divisible by 2, but I know that 378 is a composite number because 18 x 21 = 378. I didn’t use a calculator or multiply 18 and 21 by hand. I know this multiplication fact because I can see a pattern, because 18 + 3 = 21 and because SUBTRACTING 2 from a number is easy.

“What is this pattern?” you ask. Here it is. Do you see the pattern, too?

2nd Multiplication Pattern

The pattern looks like this on the multiplication table:

Basic Multiplication Table Pattern

So how did I know that 18 x 21 = 378? I knew because I can easily compute 19 x 20 = 380 in my head, and 380 – 2 = 378.

What is all the factoring information for 378?

  • 378 is a composite number.
  • Prime factorization: 378 = 2 x 3 x 3 x 3 x 7, which can be written 378 = 2 x (3^3) x 7
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 378 has exactly 16 factors.
  • Factors of 378: 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378
  • Factor pairs: 378 = 1 x 378, 2 x 189, 3 x 126, 6 x 63, 7 x 54, 9 x 42, 14 x 27, or 18 x 21
  • Taking the factor pair with the largest square number factor, we get √378 = (√9)(√42) = 3√42 ≈ 18.442

 

376 Happy Birthday, Doug

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I hope you have a wonderful day, Doug.

In our family we were the closest in age, so we often went to the same school. He was brilliant in mathematics, and I wanted to follow in his footsteps. Before I started this blog, I showed him and his wife a few Find the Factors puzzles. They were both very encouraging. We live thousands of miles away from each other, but I am grateful we were able to spend time with each other twice in 2014.

Happy birthday, Doug!

376 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

  • 376 is a composite number.
  • Prime factorization: 376 = 2 x 2 x 2 x 47, which can be written 376 = (2^3) x 47
  • 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 376 has exactly 8 factors.
  • Factors of 376: 1, 2, 4, 8, 47, 94, 188, 376
  • Factor pairs: 376 = 1 x 376, 2 x 188, 4 x 94, or 8 x 47
  • Taking the factor pair with the largest square number factor, we get √376 = (√4)(√94) = 2√94 ≈ 19.391

376 Logic

Here’s an interesting tweet about the number 376:

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373 Prime Numbers and Pythagorean Triples

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  • 373 is a prime number.
  • Prime factorization: 373 is prime.
  • The exponent of prime number 373 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 373 has exactly 2 factors.
  • Factors of 373: 1, 373
  • Factor pairs: 373 = 1 x 373
  • 373 has no square factors that allow its square root to be simplified. √373 ≈ 19.313

How do we know that 373 is a prime number? If 373 were not a prime number, then it would be divisible by at least one prime number less than or equal to √373 ≈ 19.313. Since 373 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 373 is a prime number.

  • 373 is the short leg of only one Pythagorean triple, the primitive 373, 69564, 69565
  • Prime numbers are never the longer leg, but
  • 373 is the hypotenuse of exactly one Pythagorean triple: 252, 275, 373

Here is the Odd Pythagorean triple sequence I’ve blogged about this week with the prime numbers highlighted in yellow:

Odd Primitive Primes

There are 24 odd prime numbers less than 100. The odd numbers less than 100 in this sequence produce 21 prime numbers as their hypotenuses! I think that is amazing especially since 40% of the time the hypotenuse turns out to be a composite number whose last digit is five! Here are some observations that apply to THIS sequence only:

  • When the last digit of the short leg is 3 or 7, the last digit of the hypotenuse ends in 5.
  • When the last digit of the short leg is 5, the last digit of the hypotenuse ends in 3.
  • When the last digit of the short leg is 1 or 9, the last digit of the hypotenuse ends in 1.

There are only 18 prime hypotenuses when we use about the same number of triples from this Even Primitive Triple Sequence.

Even Primitive Primes

18 primes out of 49 numbers listed is slightly less impressive than 21 primes out of 48 total numbers, but again 40% of the hypotenuses end with five and have no choice but to be composite numbers.  Here are some observations that apply to THIS sequence only:

  • When the last digit of the short leg is 4 or 6, the last digit of the hypotenuse ends in 5.
  • When the last digit of the short leg is 0, the last digit of the hypotenuse ends in 1.
  • When the last digit of the short leg is 2 or 8, the last digit of the hypotenuse ends in 7.

I had no idea that Pythagorean triples would produce so much trivia!

371 What Pythagorean Triple Comes Next?

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Could 371 be a prime number? Let’s do a quick prime number test on it.  2^371 (mod 371) = 340, not 2. Therefore, 371 is definitely a composite number. Scroll down to see its factoring information.

What Pythagorean Triple Comes Next

If you can figure out the simple pattern that these odd Pythagorean triples make, you can predict the next one in the sequence FOREVER by squaring only one number, and without ever taking a single square root! Look at the blue and red squares in the graphic above.

  • 4 + 5 = 9 = 3², and 3, 4, 5 make a Pythagorean triple.
  • 12 + 13 = 25 = 5², and 5, 12, 13 is also a Pythagorean triple.
  • 24 + 25 = 49 = 7², and 7, 24, 25 is another Pythagorean triple, and
  • 40 + 41 = 81 = 9², and 9, 40, 41 is yet another Pythagorean triple!

What Pythagorean triple comes next? The answer is found by taking the next odd number, 11, squaring it to get 121 and dividing the 121 by 2. We round down to 60 for the 2nd number and round up to 61 for the 3rd number. Thus the next Pythagorean triple would be 11, 60, 61.

No matter how many times we repeat this pattern, we always get primitive Pythagorean triples. We can put all the triples in order to make a sequence that I call the Odd Primitive Pythagorean Triple Sequence:

The Odd Primitive Pythagorean Triple Sequence

Every odd number is the short leg of at least one primitive Pythagorean triple! Here’s how I came to realize this amazing fact:

Last week I was thinking about the difference of two squares applied to integers in general

Difference of Two Squares

as I looked at this multiplication table.

The Multiplication Table

Everyone knows that the numbers in the boxes outlined in red are perfect squares, but most people do not realize that the numbers inside EVERY other colored square on this multiplication table can be expressed as the difference of two squares. The larger of those two squares will be the perfect square that is the same color.

For example, look at 5 x 11 or 55. If we count as we follow the light blue diagonal stripe from 55 to 64, we count 3 squares. That means that 55 = 64 – 3².

As I looked at the multiplication table I realized that some even numbered squares are colored and some are not, but EVERY odd number square is in color! That means that EVERY odd number on the table can be expressed as the difference of two squares in at least one way.

The light blue 9 from 1 x 9 is four squares away from the 25 that is outlined in red. 9 = 25 – 4². Since all those numbers are perfect squares, we can write 3² = 5² – 4² or the equivalent in Pythagorean Theorem form   3² + 4² = 5².

When an odd number is squared, the resulting perfect square is ALWAYS an odd number. Since all odd numbers can be written as the difference of two squares, The square of all odd numbers can be written as the difference of two squares. In other words, every odd number is a leg in a Pythagorean triple!

Let’s look at the factoring information for 371 and use it to find some Pythagorean triples.

  • 371 is a composite number.
  • Prime factorization: 371 = 7 x 53
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 371 has exactly 4 factors.
  • Factors of 371: 1, 7, 53, 371
  • Factor pairs: 371 = 1 x 371 or 7 x 53
  • 371 has no square factors that allow its square root to be simplified. √371 ≈ 19.261

371 is odd and has two odd prime factors. Here are some Pythagorean triples involving 371 as the short leg:

  • 371, 68820, 68821 (This triple is a primitive from the odd primitive sequence.)
  • 371, 1380, 1429 (A primitive, not from the odd primitive sequence. I’m also making a note that √(1380 + 1429) = 53 and √(1429 – 1380) = 7.)
  • 7, 24, 25 multiplied by 53 becomes 371, 1272, 1325
  • 53, 1404, 1405 multiplied by 7 becomes 371, 9828, 9835

371 is also the hypotenuse of a Pythagorean triple: 196, 315, 371 which is primitive 28, 45, 53 multiplied by 7.

370 and Level 3

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370 is a composite number because it is a multiple of ten. Scroll down to see its factors.

370 Puzzle

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

  • 370 is a composite number.
  • Prime factorization: 370 = 2 x 5 x 37
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 370 has exactly 8 factors.
  • Factors of 370: 1, 2, 5, 10, 37, 74, 185, 370
  • Factor pairs: 370 = 1 x 370, 2 x 185, 5 x 74, or 10 x 37
  • 370 has no square factors that allow its square root to be simplified. √370 ≈ 19.235

A Logical Approach to FIND THE FACTORS: 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.

370 Factors

368 and a Few Sequences

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368 is a composite number because it is even. Scroll down for its factoring information.

Sequences can be illustrated in pictures, but they are most often simply a list of numbers such as 23, 46, 69, 92, 115, . . . . .

Those numbers are the multiples of 23 listed in order. If we continued writing the numbers in that sequence, we would see that 368 is the 16th number listed.

We also know that 368 is an element of many other sequences even though it isn’t one of the typed numbers:

  • The sequence of all counting numbers: 1, 2, 3, 4, 5, . . . .
  • The sequence of all even counting numbers: 2, 4, 6, 8, 10, . . . .
  • The sequence of numbers that are multiples of four: 4, 8, 12, 16, 20, . . . . (This is the yellow sequence in the graphic below.)
  • The sequence of numbers that are one less than the multiples of three: 2, 5, 8, 11, 14, . . . .
  • The sequence of numbers that are one less than the multiples of nine: 8, 17, 26, 35, 44, . . . .

What's the next value

386 is NOT an element of any of the following sequences that involve even squared numbers:

  • 3, 15, 35, 63, 99, . . . . (The blue sequence in the graphic above)
  • 4, 16, 36, 64, 100, . . . .
  • 5, 17, 37, 65, 101, . . . . (The red sequence in the graphic above)

The yellow, blue, and red sequences above can be made into an array that contains only Primitive Pythagorean Triples!

Even Primitive Pythagorean Triple Sequence

Except for the 3, 4, 5 triangle that is listed first, every multiple of 4 is the short leg in a primitive Pythagorean triple! The other leg and the hypotenuse are based on a square with sides that are 1/2 the length of the shorter leg. Since the shorter leg is a multiple of four, that square length will always be an even number. Also note: 368 is divisible by 4 so 368, 33855, 33857 is a primitive that would be included in this sequence array.

The sequence contains only primitives, but it does NOT contain EVERY primitive whose short leg is a multiple of four. For example Pythagorean triple, 20, 21, 29, is not in that sequence array.

Let’s look at all the factoring information for 368:

  • 368 is a composite number.
  • Prime factorization: 368 = 2 x 2 x 2 x 2 x 23, which can be written 368 = (2^4) x 23
  • 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 368 has exactly 10 factors.
  • Factors of 368: 1, 2, 4, 8, 16, 23, 46, 92, 184, 368
  • Factor pairs: 368 = 1 x 368, 2 x 184, 4 x 92, 8 x 46, or 16 x 23
  • Taking the factor pair with the largest square number factor, we get √368 = (√16)(√23) = 4√23 ≈ 19.183

We can make a sequence array from the Pythagorean triple primitive 3, 4, 5 and its non-primitive multiples (which includes 276, 368, 460):

A Pythagorean Triple Sequence Array

We can find other Pythagorean triples that contain the number 368. Notice that 368 has several factors that are multiples of 4. They are 4, 8, 16, 92, and 184. Each one of them has its own primitive triple which can be multiplied by its factor pair partner to produce non-primitive triples that include 368 as well.

  • 3, 4, 5 multiplied by 92 is 276, 368, 460
  • 8, 15, 17 multiplied by 46 is 368, 690, 782
  • 16, 63, 65 multiplied by 23 is 368, 1449, 1495
  • 92, 2115, 2117 multiplied by 4 is 368, 8460, 8468
  • 184, 8463, 8465 multiplied by 2 is 368, 16926, 16930

In future posts I’ll write about how to find other Pythagorean triples like

  • 23, 264, 265, a primitive Pythagorean triple with an odd short leg. This primitive becomes 368, 4224, 4240 when multiplied by 16 (23’s factor pair partner).
  • 368, 465, 593, another primitive Pythagorean triple not included in the colored sequence array above and
  • 184, 513, 545, which when multiplied by 2 becomes 368, 1026, 1090.

Pythagorean triples aren’t just a bunch of “SQUARE” numbers. They are simply counting numbers that satisfy the equation a² + b² = c², and they are really COOL!

 

361 Do You See a Pattern?

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When 2^361 is divided by 361, the remainder is 116, not 2. That means that 361 is definitely a composite number. Its factors are listed at the end of this post.

361 isn’t used as often, but it is just as special as some of the numbers in the table below:

Multiplication Pattern

The pattern can also be seen along the diagonals in this ordinary multiplication table:

Multiplication Table Pattern

This pattern could be very helpful to students who are learning to multiply. I have seen plenty of students who knew 7 x 7 = 49, but couldn’t remember what 6 x 8 is.

Years after I learned the multiplication facts, I learned how to multiply binomials in an algebra class. I learned about the difference of two squares. In the example below one of the squares is n² and the other square is 1² which is equal to 1. I learned that the equation

(n-1)(n+1)

is true for ALL numbers, but nobody pointed out any practical examples to make it more meaningful. The table at the top of the page contains twelve practical examples. Let’s see how you do applying it to products of a few larger numbers.

Sometimes we find easy ways to remember certain products like
13 and 14 squared

We can use those products to help us remember other products easily by applying the difference of two squares. Try these: (Yes, you can easily do them without a calculator!)

  • 13 x 13 = 169. How much is 12 x 14?
  • 14 x 14 = 196. How much is 13 x 15?
  • 20 x 20 = 400. How much is 19 x 21?
  • If you know that the first ten powers of 2 are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, then it’s easy to remember that 16 x 16 = 256. How much is 15 x 17?
  • What if we go in the opposite direction…It isn’t too hard to multiply 18 x 20 in your head to get 360. How much is 19 x 19?
  • 22 x 20 = 440 was also easy to find. How much is 21 x 21?
  • 30 x 30 = 900. How much is 29 x 31?
  • 100 x 100 = 10,000. How much is 99 x 101?

Multiplication Pattern 2

Did you figure out what 361 has to do with this pattern? It is a perfect square just like 1, 4, 9, 16, and 25. Here is its factoring information:

  • 361 is a composite number.
  • Prime factorization: 361 = 19^2
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 361 has exactly 3 factors.
  • Factors of 361: 1, 19, 361
  • Factor pairs: 361 = 1 x 361 or 19 x 19
  • 361 is a perfect square. √361 = 19

360 What Can You Do With Fraction Circles?

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360 has more factors than any previous number. 240 and 336 held the previous record of 20 factors for each of them. How many factors do you think 360 has? Scroll down to the end of the post to find out.

360 can be evenly divided by every number from one to ten except seven, so it was a good number for the ancients to choose when they divided the circle into 360 degrees.

I bought a few fraction circles. Each 51 piece set consists of 1 whole circle as well as circles divided into 2 halves, 3 thirds, 4 fourths, 5 fifths, 6 sixths, 8 eighths, 10 tenths, and 12 twelves. What can you do with fraction circles? You can do a lot with them no matter what your age.

Art and Mathematics

The fraction circle shapes can be used just as tangram shapes to create artwork, big or small. A couple of cool symmetric designs can be found at fraction-art and fraction-circle-art. Adding rectangular fraction pieces will increase the possibilities. Here are some simple artistic designs.

Fraction Relationships

You can use fraction circle shapes to explore the relationship between fractions such as ½, ¼, and  ⅟₈;  ⅟₃, ⅟₆  and ⅟₁₂; or ½, ⅟₅ and ⅟₁₀:

Areas of Parallelograms, Trapezoids, and Circles 

The picture above shows what happens when the circle is divided into four, six, eight, ten or twelve equal wedges, and the wedges are arranged into something that resembles a parallelogram. This idea can be so easily duplicated with these fraction circles without any cutting.

Here are some good questions to ask:

  1. What happens to the top and bottom of the shape when the number of wedges increases?
  2. Sometimes the resulting shape will look like a trapezoid, and sometimes it looks more like a parallelogram. Why does that happen?

We know that the circumference of any circle is 2πr with π defined as the circumference divided by the radius. π is the same value no matter how big or small the circle is.

We can calculate the area of any of the parallelogram-like shapes or trapezoid-like shapes above. Let’s call the length of the bottom of the shape b₁ and the length of the top b₂. The area of the resulting shape is calculated: A = ½ · (b₁ + b₂) · h. Since b₁ + b₂ = 2πr, and the height equals the radius, we can write our formula for the area of a circle as A = ½ · 2πr · r = πr².

This exercise demonstrates that the area of rectangles, parallelograms, trapezoids, and circles are all related!

Introduction to Pie Charts

Pie charts are a great way to display data when we want to look at percentages of a whole. If you use fraction circles, you are limited to using only to certain percentages, but they can still make a good introduction to the subject. To make the pie chart work either the total of all the degrees will have to equal 360 or the total of all the percents will have to equal 100:

Pie Chart Pieces

After a brief introduction using the fraction circles, try Kids Zone Create a Graph. It’s really easy to use!

Exploring Perimeter and Introducing Radians in Trigonometry

The perimeter of each fraction circle piece can be calculated. If the r = 1, the circumference of the circle is 2π, and we can see an important relationship between the degrees and the perimeter of each piece.

Perimeter of Fraction Circle Pieces

What experiences have YOU had with circle fractions? Did you find them frustrating or enlightening? Personally, I like them very much, but I wish they had also been cut into ninths.

Here are some facts about the number 360:

The interior angles of every convex or concave quadrilateral total 360 degrees.

The exterior angles of every convex or concave polygon also total 360 degrees.

Here is all the factoring information about 360:

  • 360 is a composite number.
  • Prime factorization: 360 = 2 x 2 x 2 x 3 x 3 x 5, which can be written 360 = 2³·3²·5
  • The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 x 3 x 2 = 24. Therefore 360 has exactly 24 factors.
  • Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
  • Factor pairs: 360 = 1 x 360, 2 x 180, 3 x 120, 4 x 90, 5 x 72, 6 x 60, 8 x 45, 9 x 40, 10 x 36, 12 x 30, 15 x 24 or 18 x 20
  • Taking the factor pair with the largest square number factor, we get √360 = (√10)(√36) = 6√10 ≈ 18.974

Square Roots up to √352 That Can Be Simplified

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  • 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:

1-25 sqrt

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.

27-50 sqrt

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.

52-76 sqrt

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.

80-100 sqrt

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%!

104-352 sqrt

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.