Month: October 2015

644 Divisibility Rules and Level 2

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Divisibility rules 1-11 applied to the number 644:

  1. All counting numbers are divisible by 1 so 644 IS divisible by 1.
  2. 644 is even so it IS divisible by 2.
  3. 6 + 4 + 4 = 14 which is not a multiple of 3 so 644 is NOT divisible by 3.
  4. The last two digits, 44, are divisible by 4, so 644 IS divisible by 4.
  5. The last digit is not 0 or 5, so 644 is NOT divisible by 5.
  6. 644 is divisible by 2, but not by 3 so 644 is NOT divisible by 6.
  7. Breaking off the last digit, doubling it and subtracting it from the remaining digits we get: 64 – 2(4) = 56, a multiple of 7, so 644 IS divisible by 7.
  8. 6 is an even digit, and 44 is divisible by 4 but not by 8 so 644 is NOT divisible by 8.
  9. 6 + 4 + 4 = 14 which is not a multiple of 9 so 644 is NOT divisible by 9.
  10. The last digit is not 0, so 644 is NOT divisible by 10.
  11. 6 – 4 + 4 = 6 which is not a multiple of 11 so 644 is NOT a multiple of 11.

Just for the fun of it let’s try a divisibility rule for 23. Break off the last digit, multiply it by 7 and add it to the remaining digits: 64 + 7(4) = 64 + 28 = 92.

Now apply the same rule to 92: 9 + 7(2) = 9 + 14 = 23, obviously a multiple of 23, so 92 and 644 are both divisible by 23.

644 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

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  • 644 is a composite number.
  • Prime factorization: 644 = 2 x 2 x 7 x 23, which can be written 644 = (2^2) x 7 x 23
  • 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 644 has exactly 12 factors.
  • Factors of 644: 1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644
  • Factor pairs: 644 = 1 x 644, 2 x 322, 4 x 161, 7 x 92, 14 x 46, or 23 x 28
  • Taking the factor pair with the largest square number factor, we get √644 = (√4)(√161) = 2√161 ≈ 25.377155

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644 Factors

643 and Level 1

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Here’s an interesting prime factorization: 2⁶⋅3⋅643 = 123456. Did you notice its largest prime factor? Thank you OEIS.org for that fun fact about the number 643.643 PuzzlePrint the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

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  • 643 is a prime number. 641 and 643 are twin primes.
  • Prime factorization: 643 is prime.
  • The exponent of prime number 643 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 643 has exactly 2 factors.
  • Factors of 643: 1, 643
  • Factor pairs: 643 = 1 x 643
  • 643 has no square factors that allow its square root to be simplified. √643 ≈ 25.357444666.

How do we know that 643 is a prime number? If 643 were not a prime number, then it would be divisible by at least one prime number less than or equal to √643 ≈ 25.4. Since 643 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 643 is a prime number.

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643 Factors

642 Was This Venn Diagram Made Correctly?

/ ivasallay / 4 Comments

The first six multiples of 642 are 642, 1284, 1926, 2568, 3210, and 3852.

2 is a digit in each one of those numbers. OEIS.org reports that 642 is the smallest number that can make that claim.

  • 642 is a composite number.
  • Prime factorization: 642 = 2 x 3 x 107
  • 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 642 has exactly 8 factors.
  • Factors of 642: 1, 2, 3, 6, 107, 214, 321, 642
  • Factor pairs: 642 = 1 x 642, 2 x 321, 3 x 214, or 6 x 107
  • 642 has no square factors that allow its square root to be simplified. √642 ≈ 25.3377.

642 is made from 3 different even numbers. I thought it might be fun to make a Venn diagram comparing 642 with other numbers made with the same three digits. I had never made a Venn diagram on a computer before so I first tried making one in Microsoft Word, but apparently the version of Word we have doesn’t allow any writing in the parts of the Venn diagram that intersect.

I looked online for a Venn diagram maker, but didn’t use any of them for various reasons.

Finally I made a Venn diagram using different colored circles in Paint to surround information I had copied from Excel. I had to redo the work in Excel and Paint several times, but it became easier and better looking with each attempt.

Permutations of 642 and Their Factors

I attempted to show in the Venn diagram that all six numbers are divisible by 2, 3, and 6, but I’m not sure that is clear looking at the diagram. I wondered if I was even making the Venn diagram correctly in every way. Having three circles can certainly complicate the diagram. I consulted a post on Purple Math on how to solve problems using Venn diagrams , but I’m still not 100% sure I made it correctly.

I looked at Wikipedia. It showed many different types of Venn diagrams including one that sorts letters of the Greek, Latin, and Cyrillic alphabets, but the diagram wasn’t labeled.

I also saw a great Venn diagram in a post for job seekers, but it contained no data.

Being confused, what could I do? I made a completely different Venn diagram this time using Microsoft Word.

Every counting number 3 or greater is part of at least one Pythagorean triple. A number being the hypotenuse doesn’t happen as often. An even number can only be the hypotenuse if at least one of its prime factors is also an hypotenuse.

Permutations of 642-no outline

13 is a hypotenuse, and its multiple, 624, is the hypotenuse of the Pythagorean triple 240-576-624.

41 is also a hypotenuse, and its multiple, 246, is the hypotenuse of the triple 54-240-246.

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An even number being part of a primitive Pythagorean triple also only happens half the time because only numbers divisible by 4 can be part of a primitive triple.

Of the six permutations of 6-4-2, only 264 and 624 are divisible by 4, so they are the only two that are part of any primitive triples. Each of them is part of four different primitive Pythagorean triples:

  • 264-1073-1105 calculated from 2(33)(4), 33² – 4², 33² + 4²
  • 264-1927-1945 calculated from 2(44)(3), 44² – 3², 44² + 3²
  • 264-17423-17425 calculated from 2(132)(1), 132² – 1², 132² + 1²
  • 23-264-265 calculated from 12² – 11², 2(12)(11), 12² + 11²
  • 624-1457-1585 calculated from 2(39)(8), 39² – 8², 39² + 8²
  • 624-10807-10825 calculated from 2(104)(3), 104² – 3², 104² + 3²
  • 624-97343-97345 calculated from 2(312)(1), 312² – 1², 312² + 1²
  • 407-624-745 calculated from 24² – 13², 2(24)(13), 24² + 13²

I wanted the circles on the Venn diagram to be outlined so I did only one edit to them. They looked amazing in Word, but when I cut and pasted them into Paint, this is how my picture looked:

Permutations of 642

It looks like making a Venn diagram with only two circles isn’t too difficult, but adding even one more circle makes it much more complicated. Typing anything in the intersecting areas also presents a challenge no matter how many circles are used, at least in the version of Word I used.

What experiences have you had making Venn diagrams?

641 and Level 6

/ ivasallay / Leave a comment

25² + 4² = 641

641 is the hypotenuse of the primitive Pythagorean triple 200-609-641 which was calculated using 2(25)(4), 25² – 4², and 25² + 4² .

 641 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 641 is a prime number.
  • Prime factorization: 641 is prime.
  • The exponent of prime number 641 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 641 has exactly 2 factors.
  • Factors of 641: 1, 641
  • Factor pairs: 641 = 1 x 641
  • 641 has no square factors that allow its square root to be simplified. √641 ≈ 25.3179778.

How do we know that 641 is a prime number? If 641 were not a prime number, then it would be divisible by at least one prime number less than or equal to √641 ≈ 25.3. Since 641 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 641 is a prime number.

Here’s another way we know that 641 is a prime number: Since 25²+ 4² = 641, an odd number, and 25 and 4 have no common prime factors, we know that 641 is a prime number simply because it is not divisible by 5, 13, or 17.

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641 Logic

640 Fall Factor Trees and Level 5

/ ivasallay / 4 Comments

Because it is fall, and 640 has many factors, I decided to make factor trees using fall colors. Get out your rakes!

640 factor trees

There are many other possible factor trees for 640, but raking leaves can be a lot of work, so I only made two of them.

640 is the hypotenuse of the Pythagorean triple 384-512-640.

OEIS.org informs us that 640 = 16!!!!!!, but if you type 16!!!!!! into a calculator, you will get an error message as soon as you type !!.

16!!!!!! ≠ (((((16!)!)!)!)!)!

There are 6 !’s so 16!!!!!! = 16(16-6)(16-12) = 16 x 10 x 4.

Here is this week’s Level 5 puzzle:

640 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 640 is a composite number.
  • Prime factorization: 640 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 640 = (2^7) x 5
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 640 has exactly 16 factors.
  • Factors of 640: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640
  • Factor pairs: 640 = 1 x 640, 2 x 320, 4 x 160, 5 x 128, 8 x 80, 10 x 64, 16 x 40, or 20 x 32
  • Taking the factor pair with the largest square number factor, we get √640 = (√64)(√10) = 8√10 ≈ 25.298221

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640 Logic

639 and Level 4

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639 is the sum of the 20 prime numbers from 2 to 71, its largest prime factor.

639 is made from 3 numbers that are divisible by 3 so 639 is divisible by 3 AND by 9.

Puzzle #639 has a tricky clue in it, but I’m sure you can still solve it.

639 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 639 is a composite number.
  • Prime factorization: 639 = 3 x 3 x 71, which can be written 639 = (3^2) x 71
  • 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 639 has exactly 6 factors.
  • Factors of 639: 1, 3, 9, 71, 213, 639
  • Factor pairs: 639 = 1 x 639, 3 x 213, or 9 x 71
  • Taking the factor pair with the largest square number factor, we get √639 = (√9)(√71) = 3√71 ≈ 25.278449.

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639 Logic

638 and Level 3

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638 is the sum of the four prime numbers from 151 to 167.

6 – 3 + 8 = 11. Thus 638 is divisible by 11.

638 is also the hypotenuse of the Pythagorean triple 440-462-638. What is the greatest common factor of those three numbers?

638 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 638 is a composite number.
  • Prime factorization: 638 = 2 x 11 x 29
  • 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 638 has exactly 8 factors.
  • Factors of 638: 1, 2, 11, 22, 29, 58, 319, 638
  • Factor pairs: 638 = 1 x 638, 2 x 319, 11 x 58, or 22 x 29
  • 638 has no square factors that allow its square root to be simplified. √638 ≈ 25.25866.

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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. (None of the factors are greater than 12.)  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.

638 Factors

637 and Level 2

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637 is the sum of two perfect squares, 441 and 196, so it is the hypotenuse of a Pythagorean triple, namely 245-588-637. The greatest common factor of those FIVE numbers is also a perfect square. What is it?

637 is the sum of the nineteen prime numbers from 3 to 71.

637 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 637 is a composite number.
  • Prime factorization: 637 = 7 x 7 x 13, which can be written 637 = (7^2) x 13
  • 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 637 has exactly 6 factors.
  • Factors of 637: 1, 7, 13, 49, 91, 637
  • Factor pairs: 637 = 1 x 637, 7 x 91, or 13 x 49
  • Taking the factor pair with the largest square number factor, we get √637 = (√49)(√13) = 7√13 ≈ 25.2388589.

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637 Factors

636 and Level 1

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636 is the sum of the ten prime numbers from 43 to 83.

636 is also the hypotenuse of the Pythagorean triple 336-540-636. What is the greatest common factor of those three numbers?

636 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 636 is a composite number.
  • Prime factorization: 636 = 2 x 2 x 3 x 53, which can be written 636 = (2^2) x 3 x 53
  • 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 636 has exactly 12 factors.
  • Factors of 636: 1, 2, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636
  • Factor pairs: 636 = 1 x 636, 2 x 318, 3 x 212, 4 x 159, 6 x 106, or 12 x 53
  • Taking the factor pair with the largest square number factor, we get √636 = (√4)(√159) = 2√159 ≈ 25.21904

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636 Factors

635 Some Multiplication Facts to Know Backwards and Forwards

/ ivasallay / 5 Comments

Something amazing happens in the multiplication table when an even number is multiplied by 6.

Something just as amazing (but backwards) happens in the multiplication table when a multiple of 3 is multiplied by 7.

Here is a graphic that may be helpful for students memorizing some of those 6 and 7 multiplication facts. It is meant to be read from left to right and contains some cool coincidences:

2-4-6-8 Number Facts

Knowing those 6 and 7 multiplication facts will help any student know the multiplication table backwards and forwards!

While it isn’t necessary to memorize the following number facts, some of the patterns in the graphic above continue with them:

  • 10 + 5 = 15, and 15 x 7 = 105
  • 12 + 6 = 18, and 18 x 7 = 126
  • 14 + 7 = 21, and 21 x 7 = 147
  • 16 + 8 = 24, and 24 x 7 = 168
  • 18 + 9 = 27, and 27 x 7 = 189

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What makes 635 an interesting number?

635 is the sum of the nine prime numbers from 53 to 89.

It is also the sum of the thirteen prime numbers from 23 to 73.

635 can be written as the sum of 5 consecutive numbers:

125 + 126 + 127 + 128 + 129 = 635.

635 is a part of exactly 4 Pythagorean triples. Which factors of 635 are greatest common factors for the non-primitive triples?

  • 635 is the hypotenuse of the Pythagorean triple 381-508-635
  • 635 is the short leg in the triple 635-1524-1650
  • 635 is the short leg in the triple 635-40320-40325
  • 635 is the short leg in the primitive triple 635-201612-201613

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  • 635 is a composite number.
  • Prime factorization: 635 = 5 x 127
  • 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 635 has exactly 4 factors.
  • Factors of 635: 1, 5, 127, 635
  • Factor pairs: 635 = 1 x 635 or 5 x 127
  • 635 has no square factors that allow its square root to be simplified. √635 ≈ 25.199206.