factor pairs

621 and Level 6

/ ivasallay / Leave a comment

Since 23 x 27 = 621, and both of those factors are 2 away from their average, 25, we know that 621 is only 2^2 numbers away from 25^2. Mathematically we write (25-2)(25+2 ) = (25^2) – (2^2) or 23 x 27 = 625 – 4 = 621.

621 is sandwiched between 595 and 630, the 34th and the 35th triangular numbers. It can be written as the sum of consecutive numbers 7 different ways. The numbers in bold are in the exact middle of each sum. What is the relationship between the numbers in bold and the amount of numbers in the sums?

  1. 310 + 311 = 621 ( 2 consecutive numbers because 621 is divisible by 1, but not by 2)
  2. 206 + 207 + 208 = 621 (3 consecutive numbers because it is divisible by 3)
  3. 101 + 102 + 103 + 104 + 105 + 106 + 107 = 621 (6 consecutive numbers because it is divisible by 3, but not by 2)
  4. 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 = 621 (9 consecutive numbers because it is divisible by 9)
  5. 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 = 621 (18 consecutive numbers because it is divisible by 9, but not by 2)
  6. 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 = 621 (23 consecutive numbers because it is divisible by 23)
  7. 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 = 621 (27 consecutive numbers because it is divisible by 27)

621 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 621 is a composite number.
  • Prime factorization: 621 = 3 x 3 x 3 x 23, which can be written 621 = (3^3) x 23
  • 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 621 has exactly 8 factors.
  • Factors of 621: 1, 3, 9, 23, 27, 69, 207, 621
  • Factor pairs: 621 = 1 x 621, 3 x 207, 9 x 69, or 23 x 27
  • Taking the factor pair with the largest square number factor, we get √621 = (√9)(√69) = 3√69 ≈ 24.91987.

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621 Logic

620 and Level 5

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620 is the sum of the four prime numbers from 149 to 163, and it is also the sum of the eight prime numbers from 61 to 97.

620 is the hypotenuse of the Pythagorean triple 372-496-620. What is the greatest common factor of those three numbers?

Today’s puzzle may be a little more difficult than most Level 5 puzzles, but go ahead, embrace the challenge!

620 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 620 is a composite number.
  • Prime factorization: 620 = 2 x 2 x 5 x 31, which can be written 620 = (2^2) x 5 x 31
  • 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 620 has exactly 12 factors.
  • Factors of 620: 1, 2, 4, 5, 10, 20, 31, 62, 124, 155, 310, 620
  • Factor pairs: 620 = 1 x 620, 2 x 310, 4 x 155, 5 x 124, 10 x 62, or 20 x 31
  • Taking the factor pair with the largest square number factor, we get √620 = (√4)(√155) = 2√155 ≈ 24.899799

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620 Logic

 

619 and Level 4

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619  is the 12th strobogrammatic number because it looks like the same number when it is turned upside down. The first 4 strobogrammatic prime numbers are 11, 101, 181, and 619.

619 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 619 is a prime number.
  • Prime factorization: 619 is prime.
  • The exponent of prime number 619 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 619 has exactly 2 factors.
  • Factors of 619: 1, 619
  • Factor pairs: 619 = 1 x 619
  • 619 has no square factors that allow its square root to be simplified. √619 ≈ 24.87971.

How do we know that 619 is a prime number? If 619 were not a prime number, then it would be divisible by at least one prime number less than or equal to √619 ≈ 24.9. Since 619 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 619 is a prime number.

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619 Logic

 

618 and Level 3

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618 is even so it is obviously divisible by 2. Is it divisible by 3?

Yes, 618 has the same digits as 16 & 8. Since 16 is 8 doubled, 618 is divisible by 3.

618  is the sum of consecutive prime numbers 307 and 311.

618 can be written as the sum of 4 consecutive numbers because 618 is greater than 6 and is divisible by 2, but not by 4. Thus 153 + 154 + 155 + 156 = 618.

618 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 618 is a composite number.
  • Prime factorization: 618 = 2 x 3 x 103
  • 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 618 has exactly 8 factors.
  • Factors of 618: 1, 2, 3, 6, 103, 206, 309, 618
  • Factor pairs: 618 = 1 x 618, 2 x 309, 3 x 206, or 6 x 103
  • 618 has no square factors that allow its square root to be simplified. √618 ≈ 24.8596.

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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.

618 Factors

616 and Level 2

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The Padovan sequence produces a lovely spiral of equilateral triangles similar to the spiral made from golden rectangles and the Fibonacci sequence.

The Padovan sequence begins with the following numbers: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081 …

The first two numbers in the Fibonacci sequence are both 1’s. After that a number, n, in the Fibonacci sequence is found by adding together (n-2) and (n-1).

The first three numbers in the Padovan sequence are all 1’s. After that a number, n, in the Padovan sequence is found by adding together (n-3) and (n-2), and 616 is one of those numbers.

616 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 616 is a composite number.
  • Prime factorization: 616 = 2 x 2 x 2 x 7 x 11, which can be written 616 = (2^3) x 7 x 11
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 616 has exactly 16 factors.
  • Factors of 616: 1, 2, 4, 7, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616
  • Factor pairs: 616 = 1 x 616, 2 x 308, 4 x 154, 7 x 88, 8 x 77, 11 x 56, 14 x 44, or 22 x 28
  • Taking the factor pair with the largest square number factor, we get √616 = (√4)(√154) = 2√154 ≈ 24.819347.

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616 Factors

 

615 and Level 1

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615 is the hypotenuse of four Pythagorean triples. Can you find the greatest common factor for each triple?

  • 135-600-615
  • 252-561-615
  • 369-492-615
  • 399-468-615

615 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 615 is a composite number.
  • Prime factorization: 615 = 3 x 5 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 615 has exactly 8 factors.
  • Factors of 615: 1, 3, 5, 15, 41, 123, 205, 615
  • Factor pairs: 615 = 1 x 615, 3 x 205, 5 x 123, or 15 x 41
  • 615 has no square factors that allow its square root to be simplified. √615 ≈ 24.79919.

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615 Factors

614 and Level 6

/ ivasallay / 1 Comment

I learned from OEIS.org that 614 is the smallest integer that can be expressed as the sum of 3 squares 9 different ways. I decided to see if I could find those 9 ways. Here they are:

614 is the sum of 3 squares 9 different ways

614 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 614 is a composite number.
  • Prime factorization: 614 = 2 x 307
  • 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 614 has exactly 4 factors.
  • Factors of 614: 1, 2, 307, 614
  • Factor pairs: 614 = 1 x 614 or 2 x 307
  • 614 has no square factors that allow its square root to be simplified. √614 ≈ 24.77902

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614 Logic

612 and Level 5

/ ivasallay / 2 Comments

612 is the hypotenuse of the Pythagorean triple 288-540-612. Which factor of 612 is the greatest common factor of those three numbers?

612 = 17 x 36, which is 17 x 18 x 2, and that is exactly four times the formula of the 17th triangular number. Thus . . .

612 = 4(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17)

612 can be expressed as the sum of consecutive counting numbers in several ways:

  • 612 = 203 + 204 + 205 (3 consecutive numbers because it is divisible by 3)
  • 612 = 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 (eight consecutive numbers because it is divisible by 4, but not 8)
  • 612 = 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 (9 consecutive numbers because it is divisible by 9)
  • 612 = 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 (17 consecutive numbers because it is divisible by 17)
  • 612 = 14 + 15 + 16 + 17 + . . . . + 34 + 35 + 36 + 37 (24 consecutive numbers because 612 is divisible by 12, but not by 24)

What is a relationship between the numbers in bold print and the number 612?

612 is also the sum of the twelve prime numbers from 29 to 73.

612 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 612 is a composite number.
  • Prime factorization: 612 = 2 x 2 x 3 x 3 x 17, which can be written 612 = (2^2) x (3^2) x 17
  • 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 612 has exactly 18 factors.
  • Factors of 612: 1, 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204, 306, 612
  • Factor pairs: 612 = 1 x 612, 2 x 306, 3 x 204, 4 x 153, 6 x 102, 9 x 68, 12 x 51, 17 x 36 or 18 x 34
  • Taking the factor pair with the largest square number factor, we get √612 = (√36)(√17) = 6√17 ≈ 24.73863

Although I prefer using a modified cake method to find square roots, most people prefer factor trees. If you use a factor tree, I suggest you still look for easy-to-detect perfect square factors (100, 4, 9, 25) so that the most common duplicate prime factors are together:

612 Factor tree

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612 Logic

611 and Level 4

/ ivasallay / 4 Comments

611 is the sum of 13 consecutive numbers:

41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 = 611

611 is the sum of 13 consecutive odd numbers:

35 + 37 + 39 + 41 + 43 + 45 + 47 + 49 + 51 + 53 + 55 + 57 + 59 = 611

611 is the sum of the fifteen prime numbers from 13 to 71, a set of numbers that includes all the prime factors of 611.

611 is also the hypotenuse of the Pythagorean triple 235-564-611. What is the greatest common factor of those three numbers?

If you average 13 and 47 (two odd numbers in a factor pair of 611), you get 30 which is 17 numbers away from either factor. That means that (30^2) – (17^2) = 611.

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Today’s puzzle starts off easy enough, but it might get a bit tricky to complete.

611 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 611 is a composite number.
  • Prime factorization: 611 = 13 x 47
  • 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 611 has exactly 4 factors.
  • Factors of 611: 1, 13, 47, 611
  • Factor pairs: 611 = 1 x 611 or 13 x 47
  • 611 has no square factors that allow its square root to be simplified. √611 ≈ 24.718414.

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611 Logic

610 and Level 3

/ ivasallay / Leave a comment

The first 17 Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597. Notice that 610 is the 15th Fibonacci number.

There is a fascinating relationship between some of the Fibonacci numbers and some of the Markov numbers. 610 is the 12th Markov number. Get out your calculator and satisfy yourself that the following two Diophantine equations involving Fibonacci/Markov numbers are true:

1² + 233² + 610² = 3(1)(233)(610)

1² + 610² + 1597² = 3(1)(610)(1597)

Here is a fascinating fact I learned from twitter:

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Starting with 5, every other Fibonacci number would be 5, 13, 34, 89, 233, 610, 1597, . . .

610 is on that list. What could be the integer sides of a right triangle with 610 as the hypotenuse?

There are actually FOUR such triangles, namely. . .

  • 110-600-610
  • 272-546-610
  • 414-448-610
  • 366-488-610

None of those are primitives, but it is a great list nonetheless!

610 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 610 is a composite number.
  • Prime factorization: 610 = 2 x 5 x 61
  • 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 610 has exactly 8 factors.
  • Factors of 610: 1, 2, 5, 10, 61, 122, 305, 610
  • Factor pairs: 610 = 1 x 610, 2 x 305, 5 x 122, or 10 x 61
  • 610 has no square factors that allow its square root to be simplified. √610 ≈ 24.698178.

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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. 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.

610 Factors