Mathematics

269 and Five More Consecutive Square Roots

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  • 269 is a prime number.
  • Prime factorization: 269 is prime.
  • The exponent of prime number 269 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 269 has exactly 2 factors.
  • Factors of 269: 1, 269
  • Factor pairs: 269 = 1 x 269
  • 269 has no square factors that allow its square root to be simplified. √269 ≈ 16.401

How do we know that 269 is a prime number? If 269 were not a prime number, then it would be divisible by at least one prime number less than or equal to √269 ≈ 16.401. Since 269 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 269 is a prime number.

As I have previously written, 844, 845, 846, 847, and 848 are the smallest FIVE consecutive numbers whose square roots can be simplified. Here are the second smallest FIVE with the same property.

1680 square roots

The first number in the second set, 1680, equals 2 x 840 which is very close to the first number in the first set. Will strings of five consecutive numbers with reducible square roots occur about once every 850 numbers?

We can find the number of factors for these numbers by examining their prime factorizations.

1680 prime factorization

The number of factors for each of the integers in this second set ranges from 3 to 40. Only two of the integers have the same number of factors. Finding another string of four or more numbers that have reducible square roots as well as the same number of factors may be difficult.

266 Why 8 Consecutive Numbers with 6 Factors is Impossible

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  • 266 is a composite number.
  • Prime factorization: 266 = 2 x 7 x 19
  • 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 266 has 8 factors.
  • Factors of 266: 1, 2, 7, 14, 19, 38, 133, 266
  • Factor pairs: 265 = 1 x 266, 2 x 133, 7 x 38, or 14 x 19
  • 266 has no square factors that allow its square root to be simplified. √266 ≈ 16.3095

242, 243, 244, and 245 are the smallest four consecutive numbers that have the same number of factors. Each of them has exactly six factors, and as a result each one of their square roots can be simplified. Is it possible to have a longer string of consecutive numbers with exactly six factors? I don’t know yet. It seems reasonable that it could happen, so I am on the lookout for five, six, or seven consecutive numbers that have exactly six factors.

I won’t bother looking for a string of eight or more numbers with six factors. I already know that would be impossible. Here’s an example illustrating why:

844 - 848 prime factorization

846 ruins the run because it has an additional prime factor that doubles the number of factors that 846 has in all. Further down the number line that problem might be overcome in a different set of consecutive numbers.

However, the problem with 848 is a recurring problem that will never be overcome. 848 is divisible by 8, as is every eighth number. The prime factorization of numbers that are divisible by eight must contain a power of two that is greater than or equal to three. Its number of factors calculation would have to be at least (3 + 1)(1 + 1) = 4 x 2 = 8. (The ONLY number divisible by 8 that has exactly 6 factors is 32.)

Even though the numbers from 844 to 848 don’t have the same number of factors, they still have a distinction. They are the smallest five consecutive numbers whose square roots can be simplified!

844, 845, 846, 847, 848

263 How and Why You Should Show Your Work

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  • 263 is a prime number.
  • Prime factorization: 263 is prime.
  • The exponent of prime number 263 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 263 has exactly 2 factors.
  • Factors of 263: 1, 263
  • Factor pairs: 263 = 1 x 263
  • 263 has no square factors that allow its square root to be simplified. √263 ≈ 16.217

How do we know that 263 is a prime number? If 263 were not a prime number, then it would be divisible by at least one prime number less than or equal to √263 ≈ 16.217. Since 263 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 263 is a prime number.

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Joseph Nebus reads hundreds of comics every day as he looks for ones with a mathematical theme. He regularly shares these finds on his blog and writes a short explanation about the mathematics mentioned in the comics. I especially loved his reading-the-comics-october-14-2014 edition.

Why mathematics students should show their work is clearly explained under Jeff Mallet’s Frazz (October 12) comic strip.  Joseph Nebus basically gives two reasons to show work. The first reason has probably been stated by teachers thousands of times, but the second is truly an inspiration, and I highly recommend teachers and students alike read it!

How should the work be shown? This You-tube video does a very good job showing how to show work and make that work as readable as possible.

260 Some Thoughts on Those Four Consecutive Numbers

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  • 260 is a composite number.
  • Prime factorization: 260 = 2 x 2 x 5 x 13, which can be written 260 = (2^2) x 5 x 13
  • 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 260 has 12 factors.
  • Factors of 260: 1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260
  • Factor pairs: 260 = 1 x 260, 2 x 130, 4 x 65, 5 x 52, 10 x 26, or 13 x 20
  • Taking the factor pair with the largest square number factor, we get √260 = (√4)(√65) = 2√65 ≈ 16.125

Recently I wrote about the smallest four-consecutive-numbers whose square roots could all be simplified. The same numbers were also the smallest four consecutive numbers to have the same number of factors.

4 numbers; 6 factors

Each of those numbers had 6 factors, and guess what, ANY number with exactly 6 factors can have its square root simplified. The prime factorization of ANY number with exactly 6 factors can be expressed in one of the three following ways:

6 factors

Since numbers with six factors always have a prime factor raised to a power greater than one, they can always have their square roots simplified. The fact that those four consecutive numbers have the same number of factors makes them extraordinary; that they all can have their square roots simplified is merely the natural consequence of that extraordinary fact.

252 How likely can this square root be simplified?

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  • 252 is a composite number.
  • Prime factorization: 252 = 2 x 2 x 3 x 3 x 7, which can be written 252 = (2^2) x (3^2) 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 252 has 18 factors.
  • Factors of 252: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252
  • Factor pairs: 252 = 1 x 252, 2 x 126, 3 x 84, 4 x 63, 6 x 42, 7 x 36, 9 x 28, 12 x 21, or 14 x 18
  • Taking the factor pair with the largest square number factor, we get √252 = (√7)(√36) = 6√7 ≈ 15.875

The square root of a whole number can be simplified if it has a square number factor. How likely is that condition met by any random whole number?

4 is 2 x 2 and therefore a square number.  1 out of every four whole numbers (or 25%) is divisible by 4

3^2 = 9. Likewise 1 out of every nine whole numbers is divisible by square number 9 (about 11.1%).

Some numbers, like 252, are divisible by both 4 and 9. (1 out of every 36 numbers are divisible by both 4 and 9.)

 

1 third

Thus 1/3 of all whole numbers are divisible by 4, 9 or both.

That means that 2/3 of the numbers in the set of all whole numbers are NOT divisible by 4, 9 or both. It is often easier to compute the probability of something NOT happening and then subtract that fraction from 1 to determine the probability of something actually happening. The probability a number is NOT divisible by 4 is 3/4 while the probability a whole number is NOT divisible by 9 is 8/9. We get the same result either way.

1 - 2 thirds

1/3 of all whole numbers (about 33.3%) are divisible by either 4 or 9! That fact is very cool because it is so easy to tell if a number is divisible by 4 or 9: If the last 2 digits of a number is divisible by 4, the entire number is divisible by 4 and if the sum of the digits of a whole number is divisible by 9, that whole number is divisible by 9.

It is also very easy to tell if a number is divisible by 5 x 5 or 25. If the last two digits of the number are 25, 50, 75 or 00, then it is divisible by 25. Let’s compute how likely it is that the square root of a number can be simplified because that number is divisible by 4, 9, or 25.

9 twenty-fifths

Thus 36% of all whole numbers are divisible by 4, 9, or 25 and therefore have square roots that can be simplified! It is not as easy to tell if a number is divisible by 49, 121, 169, or any other number that is the perfect square of a prime number. The percentage of numbers that are divisible by these other perfect squares doesn’t go up much more either. Consider this infinite product subtracted from 1:

nearing 40%

When I’ve computed the partial product up to 3480/(59 x 59) and subtracted it from 1, the probability only increased to 39.010%. I used excel to compute the probability of a number being divisible by a square factor up to 1,495,729 (which is 1223^2) and it is only 39.201%. There isn’t much change in the percentage between the 17th prime number (59) and the 200th prime number (1223).

As n gets larger (n^2 -1)/(n^2) gets closer and closer to 1. I conclude that the probability that a random whole number can have its square root simplified is about 40%.

246 and Level 1

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  • 246 is a composite number.
  • Prime factorization: 246 = 2 x 3 x 41
  • 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 246 has 8 factors.
  • Factors of 246: 1, 2, 3, 6, 41, 82, 123, 246
  • Factor pairs: 246 = 1 x 246, 2 x 123, 3 x 82, or 6 x 41
  • 246 has no square factors so its square root cannot be simplified. √246 ≈ 15.684

 2014-39 Level 1

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-29

2014-39 Level 1 Factors

245 – The Last of Four Consecutive Numbers

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  • 245 is a composite number.
  • Prime factorization: 245 = 5 x 7 x 7, which can be written 245 = 5 x (7^2)
  • The exponents in the prime factorization are 1 and 2. Adding one  to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3 = 6. Therefore 245 has 6 factors.
  • Factors of 245: 1, 5, 7, 35, 49, 245
  • Factor pairs: 245 = 1 x 245, 5 x 49, or 7 x 35
  • Taking the factor pair with the largest square number factor, we get √245 = (√5)(√49) = 7√5 ≈ 15.652

Square roots 242 - 245

 

I was surprised when I noticed that the square roots of these 4 consecutive numbers – 242, 243, 244, and 245 could all be simplified.

The square root of a whole number can only be simplified if that whole number has a square number as one of its factors. All four of these numbers meet that condition, and they are the first four consecutive numbers to do so.

For numbers less than or equal to 240, there are only 3 sets of 3 consecutive square roots that can be simplified.

  • √48 = 4√3
  • √49 = 7
  • √50 = 5√2
  • √98 = 7√2
  • √99 = 3√11
  • √100 = 10
  • √124 = 2√31
  • √125 = 5√5
  • √126 = 3√14

242, 243, 244, and 245 also have another distinction. They each have exactly 6 factors and are the smallest consecutive four numbers to have the same number of factors.

242 and Level 4

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  • 242 is a composite number.
  • Prime factorization: 242 = 2 x 11 x 11, which can be written 2 x (11^2)
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3 = 6. Therefore 242 has 6 factors.
  • Factors of 242: 1, 2, 11, 22, 121, 242
  • Factor pairs: 242 = 1 x 242, 2 x 121, or 11 x 22
  • Taking the factor pair with the largest square number factor, we get √242 = (√2)(√121) = 11√2 ≈ 15.556

2014-38 Level 4

Print the puzzles or type the factors on this excel file: 12 Factors 2014-09-22

2014-38 Level 4 Logic

A Forest of 240 Factor Trees

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Factor Trees for 240:

Because 240 has so many factors, it is possible to make MANY different factor trees that create a forest of 240 factor trees. This post only contains eleven of those many possibilities. The two trees below demonstrate different permutations that can be made from the same basic tree. The mirror images of both, as well as mirror images of parts of either tree, would be other permutations.

240 Factor Trees

 

A good way to make a factor tree for a composite number is to begin with one of its factor pairs and then make factor trees for the composite numbers in that factor pair.

240 Factor Trees 1 - 3

In this first set of three factor trees we can also see the factor trees for 120, 80, 4, & 60.

240 Factor Trees 4 - 6

These three factor trees also include factor trees for 48, 6, 40, 8, and 30.

240 Factor Trees 7 - 9

Finally, these three factor trees also include factor trees for 10, 24, 12, 20, 15, and 16.

This forest of 240 factor trees is dedicated to Joseph Nebus. Read the comments to his post, You might also like, because, I don’t know why, to discover why I was inspired to create images of parts of this forest.

Factors of 240:

  • 240 is a composite number.
  • Prime factorization: 240 = 2 x 2 x 2 x 2 x 3 x 5, which can be written 2⁴ x 3 x 5
  • 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 x 2 x 2 = 20. Therefore 240 has 20 factors.
  • Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
  • Factor pairs: 240 = 1 x 240, 2 x 120, 3 x 80, 4 x 60, 5 x 48, 6 x 40, 8 x 30, 10 x 24, 12 x 20, or 15 x 16
  • Taking the factor pair with the largest square number factor, we get √240 = (√16)(√15) = 4√15 ≈ 15.492

Sum-Difference Puzzle:

60 has six factor pairs. One of those factor pairs adds up to 17, and a different one subtracts to 17. Can you find those factor pairs to solve the first puzzle below?

240 has ten factor pairs. One of them adds up to 34, and another one subtracts to 34. If you can identify those factor pairs, then you can solve the second puzzle.

The second puzzle is really just the first puzzle in disguise. Why would I say that?

Another Fact about the Number 240:

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 = 240; that’s the sum of the first 15 even numbers.

 

How Many Factors Does 224 Have, and Why Does It Have That Many?

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  • 224 is a composite number.
  • Prime factorization: 224 = 2 x 2 x 2 x 2 x 2 x 7, which can be written 224 = 2⁵ x 7
  • 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 224 has 12 factors.
  • Factors of 224: 1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112, 224
  • Factor pairs: 224 = 1 x 224, 2 x 112, 4 x 56, 7 x 32, 8 x 28, or 14 x 16
  • Taking the factor pair with the largest square number factor, we get √224 = (√16)( √14) = 4√14 ≈ 14.967

It may seem like a little mathemagic to state that 224 has 12 factors because the exponents in its prime factorization are 5 and 1 and because 6 x 2 = 12. If we look at the factors of 224 in a different way, all of the mathemagician’s secrets will be revealed.

First it must be understood that any number (except zero) raised to the zeroth power is equal to one.

Since the prime factorization of 224 is 2⁵ x 7¹, every one of its factors can be written as the product of certain powers of 2 and 7:

factors of 224

Counting all the factors of 224 is similar to counting the number of possible sandwiches when there are 5 different meats (and vegetarians are given the option of no meat) and one type of bread (and those counting carbs are given the option of no bread). Even though there are 5 different types of meat, there are 6 possible choices about meat, while bread or no bread makes 2 possible choices about bread. The fundamental counting principle states that we can count the total number of possible sandwiches by multiplying together the number of possible choices of meat and bread. In this case that is 6 x 2 = 12.

(We are allowing people to chose to have no meat and/or no bread. Usually when the fundamental counting principle is used we don’t make that allowance.  Allowing the sandwiches to be made without meat and/or bread means we have to add 1 to the 5 types of meat and 1 to the 1 type of bread before we multiply those numbers together, giving us (5 + 1)(1 + 1) or 6 x 2. If we did not allow anyone to choose to leave out the meat and/or bread we would not add 1.)

Adding even one more number in the prime factorization or adding even one more ingredient to the sandwich can make counting the factors or sandwiches more tedious:

If we also allow 3 different types of lettuce (or no lettuce for those who don’t want any), the problem becomes a little more complicated, but the fundamental counting principle makes it easy to find the number of possible types of sandwiches. Since we now have 4 lettuce choices, we simply multiply the previous information by that number: (6 x 2) x 4 = 12 x 4 = 48. Finding the number of possible sandwiches is easy, but listing all 48 sandwiches might make some people lose their appetite.

Just as it worked for finding the number of possible sandwiches, the fundamental counting principle also helps us know the number of factors a whole number has. When each number in a factor pair is written using a modified version of its prime factorization (to allow us to write a factor to the zeroth power), then we can clearly see how to make (2^5) x (7 ^1) with every factor pair.

factor pairs for 224

 

If the prime factorization of a number were (2^5) x (7^1) x (11^3), it would have (5+1)(1+1)(3+1) = 6 x 2 x 4 = 48 factors. Finding the number of factors may be much easier than listing all of them, but knowing how many factors there are helps us make sure we don’t list too few or too many.

When a number is prime such as 7, it has no prime factorization, but 7 can be represented as (7^1). Its exponent is 1; (1 + 1) = 2, and its two factors can be listed as (7^0) and (7^1) or 1 and 7. All prime numbers have exactly 2 factors.

Factor pairs obviously come in 2’s, so will the number of factors a number has always be an even number?

We know from number theory when we multiply an even number by any other number, we always get an even number. To get an odd number as the product, EVERY number that is multiplied together would have to be odd. Therefore, the only way to get an odd number when we use the fundamental counting principle is if all the numbers being multiplied together are odd. In the case of counting the number of factors of a number, we only get an odd number of factors when the number being factored is a perfect square. That is the only way ALL of the exponents in its prime factorization are even numbers, and those even exponents plus 1 will ALL be odd.

For example, if the prime factorization were (2^6)(3^2), then the number of factors would be (6 + 1)(2 + 1) = 7 x 3 = 21, a odd number. One of its factor pairs could be written (2^3)(3^1) x (2^3)(3^1), but is normally listed as 24 x 24. So when we list the factor pairs there will be 11 pairs, but when we list the 21 factors of 576, we only list 24 once. Fortunately teachers and textbooks rarely choose numbers with that many factors.

When factoring numbers, making sandwiches, or making many other choices, the fundamental counting principle gives the number of all the possible outcomes and makes it easier not to overlook any possibilities.