Prime factorization

253 and Level 1

/ ivasallay / Leave a comment

  • 253 is a composite number.
  • Prime factorization: 253 = 11 x 23
  • 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 253 has 4 factors.
  • Factors of 253: 1, 11, 23, 253
  • Factor pairs: 253 = 1 x 253 or 11 x 23
  • 253 has no square factors that allow its square root to be simplified. √253 ≈ 15.906

2014-40 Level 1

Print the puzzles or type the factors on this excel file: 12 Factors 2014-10-06

2014-40 Level 1 Factors

252 How likely can this square root be simplified?

/ ivasallay / 2 Comments

  • 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%.

251 and Level 6

/ ivasallay / Leave a comment

  • 251 is a prime number.
  • Prime factorization: 251 is prime.
  • The exponent of prime number 251 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 251 has exactly 2 factors.
  • Factors of 251: 1, 251
  • Factor pairs: 251 = 1 x 251
  • 251 has no square factors so its square root cannot be simplified. √251 ≈ 15.843

How do we know that 251 is a prime number? If 251 were not a prime number, then it would be divisible by at least one prime number less than or equal to √251 ≈ 15.843. Since 251 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 251 is a prime number.

2014-39 Level 6

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

2014-39 Level 6 Logic

250 and Level 5

/ ivasallay / Leave a comment

  • 250 is a composite number.
  • Prime factorization: 250 = 2 x 5 x 5 x 5, which can be written 250 = 2 x (5^3)
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 x 4 = 8. Therefore 250 has 8 factors.
  • Factors of 250: 1, 2, 5, 10, 25, 50, 125, 250
  • Factor pairs: 250 = 1 x 250, 2 x 125, 5 x 50, or 10 x 25
  • Taking the factor pair with the largest square number factor, we get √250 = (√10)(√25) = 5√10 ≈ 15.811

2014-39 Level 5

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

2014-39 Level 5 Logic

249 and Level 4

/ ivasallay / Leave a comment

  • 249 is a composite number.
  • Prime factorization: 249 = 3 x 83
  • 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 249 has 4 factors.
  • Factors of 249: 1, 3, 83, 249
  • Factor pairs: 249 = 1 x 249 or 3 x 83
  • 249 has no square factors so its square root cannot be simplified. √249 ≈ 15.7797

2014-39 Level 4

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

2014-39 Level 4 Logic

248 and Level 3

/ ivasallay / Leave a comment

  • 248 is a composite number.
  • Prime factorization: 248 = 2 x 2 x 2 x 31, which can be written 248 = (2^3) x 31
  • 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 248 has 8 factors.
  • Factors of 248: 1, 2, 4, 8, 31, 62, 124, 248
  • Factor pairs: 248 = 1 x 248, 2 x 124, 4 x 62, or 8 x 31
  • Taking the factor pair with the largest square number factor, we get √248 = (√4)(√62) = 2√62 ≈ 15.748

2014-39 Level 3

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

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.

2014-39 Level 3 Factors

247 and Level 2

/ ivasallay / Leave a comment

  • 247 is a composite number.
  • Prime factorization: 247 = 13 x 19
  • 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 247 has 4 factors.
  • Factors of 247: 1, 13, 19, 247
  • Factor pairs: 247 = 1 x 247 or 13 x 19
  • 247 has no square factors so its square root cannot be simplified. √247 ≈ 15.716

2014-39 Level 2

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

2014-39 Level 2 Factors

246 and Level 1

/ ivasallay / Leave a comment

  • 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

/ ivasallay / 3 Comments

  • 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.

244 and Level 6

/ ivasallay / Leave a comment

  • 244 is a composite number.
  • Prime factorization: 244 = 2 x 2 x 61, which can be written 244 = (2^2) x 61
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 244 has 6 factors.
  • Factors of 244: 1, 2, 4, 61, 122, 244
  • Factor pairs: 244 = 1 x 244, 2 x 122, 4 x 61
  • Taking the factor pair with the largest square number factor, we get √244 = (√4)(√61) = 2√61 ≈ 15.6205

2014-38 Level 6

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

2014-38 Level 6 Logic