Factors

679 and Level 6

/ ivasallay / Leave a comment

679 is the sum of the three primes from 223 to 229 and the sum of the nine primes from 59 to 97.

Since 97 is one of its factors, 679 is the hypotenuse of the Pythagorean triple 455-504-679. What is the greatest common factor of those three numbers?

679 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 679 is a composite number.
  • Prime factorization: 679 = 7 x 97
  • 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 679 has exactly 4 factors.
  • Factors of 679: 1, 7, 97, 679
  • Factor pairs: 679 = 1 x 679 or 7 x 97
  • 679 has no square factors that allow its square root to be simplified. √679 ≈ 26.057628.

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679 Logic

678 and Level 5

/ ivasallay / Leave a comment

678 is made from 3 consecutive numbers so it is divisible by 3. The middle number of those 3 consecutive numbers is not divisible by 3, so 678 is NOT divisible by 9.

678 is the hypotenuse of  Pythagorean triple 90-672-678. What is the greatest common factor of those 3 numbers?

678 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 678 is a composite number.
  • Prime factorization: 678 = 2 x 3 x 113
  • 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 678 has exactly 8 factors.
  • Factors of 678: 1, 2, 3, 6, 113, 226, 339, 678
  • Factor pairs: 678 = 1 x 678, 2 x 339, 3 x 226, or 6 x 113
  • 678 has no square factors that allow its square root to be simplified. √678 ≈ 26.038433.

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678 Logic

676 and Level 3

/ ivasallay / 2 Comments

676 is a palindrome in three consecutive different bases plus a couple of other bases:

  • 10201 in base 5; note that 1(625) + 0(125) + 2(25) + 0(5) + 1(1) = 676.
  • 676 in base 10; note that 6(100) + 7(10) + 6(1) = 676.
  • 565 in base 11; note that 5(121) + 6(11) + 5(1) = 676.
  • 484 in base 12; note that 4(144) + 8(12) + 4(1) = 676.
  • 121 in base 25; note that 1(625) + 2(25) + 1(1) = 676.

Speaking of palindromes, OEIS.org states that 676 is the smallest perfect square palindrome whose square root is not also a palindrome. (Palindromic perfect squares less than 676 are 1, 4, 9, 121, and 484.)

Since 13 and 169 are two of its factors, 676 is also the hypotenuse of Pythagorean triples 476-480-676 and 260-624-676. What is the greatest common factor of each triple?

676 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 676 is a composite number.
  • Prime factorization: 676 = 2 x 2 x 13 x 13, which can be written 676 = (2^2) x (13^2)
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 x 3 = 9. Therefore 676 has exactly 9 factors.
  • Factors of 676: 1, 2, 4, 13, 26, 52, 169, 338, 676
  • Factor pairs: 676 = 1 x 676, 2 x 338, 4 x 169, 13 x 52, or 26 x 26
  • 676 is a perfect square. √676 = 26

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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 10.)  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.

676 Factors

675 and Level 2

/ ivasallay / Leave a comment

675 is made from 3 consecutive numbers so 675 is divisible by 3. Since the middle number of those consecutive numbers, 6, is divisible by 3, we know that 675 is also divisible by 9.

Since the last two digits of 675 is a multiple of 25, we know that 675 is divisible by 25.

Since 5 and 25 are two of its factors, 675 is the hypotenuse of two Pythagorean triples: 405-540-675 and 189-648-675. What is the greatest common factor of each set of three numbers?

The numbers in one of 675’s factor pairs, 25 and 27, are each exactly one number away from 26, their average. That means we are just one number away from 26².

Thus, 675 equals 26² – 1² which can be factored into (26 + 1)(26 – 1) so (26 + 1)(26 – 1) = 27 x 25 = 675.

675 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 675 is a composite number.
  • Prime factorization: 675 = 3 x 3 x 3 x 5 x 5, which can be written 675 = (3^3) x (5^2)
  • The exponents in the prime factorization are 3 and 2. Adding one to each and multiplying we get (3 + 1)(2 + 1) = 4 x 3 = 12. Therefore 675 has exactly 12 factors.
  • Factors of 675: 1, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675
  • Factor pairs: 675 = 1 x 675, 3 x 225, 5 x 135, 9 x 75, 15 x 45, or 25 x 27
  • Taking the factor pair with the largest square number factor, we get √675 = (√225)(√3) = 15√3 ≈ 25.98076.

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675 Factors

672 Is it too soon to pick out a tree?

/ ivasallay / Leave a comment

672 can make MANY factor trees.  Here I’ve pictured only a few of the possibilities, one for each of its factor pairs (excluding 1 x 672).

Is it too soon to pick out a tree?

672 Factor Trees

Every one of those trees has the prime factors of 672: 2, 2, 2, 2, 2, 3, and 7, but finding them on each tree might be a challenge because I didn’t distinguish the prime factors from the other factors. Some of those prime factors might seem like they are lost in a pile of leaves. Can you find them on each tree?

672 is the 9th number to have 24 factors. Here is a number line highlighting all nine of those numbers and the distances between them.

24 Factors Number Line

Notice the difference between 672 and the previous number with 24 factors is 12, a record low.

You might get the impression looking at the number line that numbers having 24 factors might be much more common from now on. That may be true, nevertheless, the next number after 672 to have exactly 24 factors is 756 which is 84 numbers away and well past 720 the smallest number to have 30 factors.

Indeed infinitely many numbers have 24 factors, but probably 672 is the last one that will get much attention.

The numbers in the factor pair 24 and 28 are each exactly two numbers away from 26, their average. That means we are just 2² numbers away from 26².

In other words, 672 equals 26² – 2² which can be factored into (26 + 2)(26 – 2) so (26 + 2)(26 – 2) = 28 x 24 = 672.

  • 672 is a composite number.
  • Prime factorization: 672 = 2 x 2 x 2 x 2 x 2 x 3 x 7, which can be written 672 = (2^5) x 3 x 7
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 x 2 x 2 = 24. Therefore 672 has exactly 24 factors.
  • Factors of 672: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672
  • Factor pairs: 672 = 1 x 672, 2 x 336, 3 x 224, 4 x 168, 6 x 112, 7 x 96, 8 x 84, 12 x 56, 14 x 48, 16 x 42, 21 x 32, or 24 x 28
  • Taking the factor pair with the largest square number factor, we get √672 = (√16)(√42) = 4√42 ≈ 25.92296.

671 is the Magic Sum of an 11 x 11 Magic Square

/ ivasallay / 3 Comments

6 – 7 + 1 = 0 so 671 is divisible by 11.

671 is the sum of the fifteen prime numbers from 17 to 73.

Because 61 is one of its factors, 671 is the hypotenuse of the Pythagorean triple 121-660-671. The greatest common factor of those three numbers practically jumps out at me. Does it do the same thing to you?

Best of all 671 is the magic sum of an 11 x 11 magic square. (That link from wikipedia helped me construct this square. I’ll give directions so you can do it, too!)

671 Magic Sum for 11 x 11 Magic Square

Notice how every row, column, and diagonal on the square sums to 671. The reason it is the magic sum is because the sum of all the numbers from 1 to 121 can be computed and then divided by 11 (the number of rows). Here is the equation:

  • 671 = 121 x 120/2/11

Because 11 is an odd number there are simple directions to complete the entire square:

The number 1 is located in the exact center of the top row.

Find the number 2 on the square. (It’s located on the bottom row just right of the exact center square.) Notice that the numbers 3, 4, 5, and 6 are on the same diagonal. If you imagine the diagonal wrapping around the square, you can continue to follow it for numbers 7, 8, 9, 10, and 11. We can’t put the number 12 along the same diagonal because the number 1 is already in that spot, so we put the 12 UNDER the 11 and begin working on a new diagonal.

Anytime a number already occupies a space on a diagonal, put the next number under the preceding number and continue making a new diagonal. When a diagonal reaches the edge of the square, imagine that edge is connected to the opposite edge and continue the diagonal from the opposite edge.

I found it to be the trickiest placing the numbers 67 and 68, but other than that it was rather easy to know where to put the numbers.

Notice that the difference between any smaller number and the larger number just below it is either 12 or 1.

If you have excel on your computer, click on 12 Factors 2015-11-02, select the magic square tab, and then you can make this 11 x 11 magic square yourself. As you type in numbers, the columns, rows, and diagonals will automatically keep a running sum.

Once you get the square to give the magic sum in each direction, you can try doing the same thing with the 13 x 13 magic square that I’ve included on the same page. Its magic sum is 1105 which can be also be computed:

  • 1105 = 169 x 170/22/13.

There is actually many more possible and probably more complicated 11 x 11 and 13 x 13 magic squares. I hope you enjoy making some with this easy method.

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  • 671 is a composite number.
  • Prime factorization: 671 = 11 x 61
  • 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 671 has exactly 4 factors.
  • Factors of 671: 1, 11, 61, 671
  • Factor pairs: 671 = 1 x 671 or 11 x 61
  • 671 has no square factors that allow its square root to be simplified. √671 ≈ 25.903668.

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670 and Level 5

/ ivasallay / Leave a comment

Because 5 is one of its factors, 670 is the hypotenuse of the Pythagorean triple 402-536-670. Which factor of 670 is the greatest common factor of those three numbers?

670 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

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  • 670 is a composite number.
  • Prime factorization: 670 = 2 x 5 x 67
  • 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 670 has exactly 8 factors.
  • Factors of 670: 1, 2, 5, 10, 67, 134, 335, 670
  • Factor pairs: 670 = 1 x 670, 2 x 335, 5 x 134, or 10 x 67
  • 670 has no square factors that allow its square root to be simplified. √670 ≈ 25.884358.

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670 Logic

Thank you, Ricardo, for tweeting the solution:

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669 and Level 4

/ ivasallay / Leave a comment

Since all its digits are divisible by 3, obviously 669 is divisible by 3.

222 + 223 + 224 = 669. Thus 669 is the sum of 3 consecutive numbers. Prime factor 223 is the middle number in the sum.

Also consecutive numbers 109 + 110 + 111 + 112 + 113 + 114 = 669

669 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

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  • 669 is a composite number.
  • Prime factorization: 669 = 3 x 223
  • 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 669 has exactly 4 factors.
  • Factors of 669: 1, 3, 223, 669
  • Factor pairs: 669 = 1 x 669 or 3 x 223
  • 669 has no square factors that allow its square root to be simplified. √669 ≈ 25.865034.

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669 Logic

Ricardo shows the solution here:

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668 and Level 3

/ ivasallay / Leave a comment

668 is the sum of consecutive prime numbers 331 and 337.

68 is divisible by 4 so 668 and every other number ending in 68 is divisible by 4.

80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 = 668

668 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

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  • The first ten multiples of 668 are 668, 1336, 2004, 2672, 3340, 4008, 4676, 5344, 6012, and 6680.
  • 668 is a composite number.
  • Prime factorization: 668 = 2 x 2 x 167, which can be written 668 = (2^2) x 167
  • 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 668 has exactly 6 factors.
  • Factors of 668: 1, 2, 4, 167, 334, 668
  • Factor pairs: 668 = 1 x 668, 2 x 334, or 4 x 167
  • Taking the factor pair with the largest square number factor, we get √668 = (√4)(√167) = 2√167 ≈ 25.845696.

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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.

668 Factors

667 and Level 2

/ ivasallay / Leave a comment

Since 29 is one of its factor, 667 is the hypotenuse of the Pythagorean triple 460-483-667. What is the greatest common factor of those three numbers?

The numbers in the factor pair 23 and 29 are each exactly three numbers away from 26, their average. That means we are just 3² numbers away from 26².

Indeed, 667 equals 26² – 3² which can be factored into (26 + 3)(26 – 3) so (26 + 3)(26 – 3) = 29 x 23 = 667.

667 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

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  • 667 is a composite number.
  • Prime factorization: 667 = 23 x 29
  • 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 667 has exactly 4 factors.
  • Factors of 667: 1, 23, 29, 667
  • Factor pairs: 667 = 1 x 667 or 23 x 29
  • 667 has no square factors that allow its square root to be simplified. √667 ≈ 25.826343.

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667 Factors