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

609 and Level 2

/ ivasallay / Leave a comment

609 is the hypotenuse of the Pythagorean triple 420-441-609. What is the greatest common factor of those three numbers?

21 x 29 = 609, both odd numbers from a factor pair for 609, and (29 – 21)/2 = 4. So since 4^2 = 16, we are 16 counting numbers away from a perfect square, in fact the next perfect square. Written mathematically that is (25-4)(25 +4) = (25^2) – (4^2).

609 is the 11th strobogrammatic number which means it looks like the same number upside-down because it uses only the digits 0, 1, 6, 8, and 9.

There are other numbers that don’t look the same upside-down but look the same reflected in a mirror. Check out teachfuthermaths for a fun puzzle about some of those numbers.

 609 Puzzle

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

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  • 609 is a composite number.
  • Prime factorization: 609 = 3 x 7 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 609 has exactly 8 factors.
  • Factors of 609: 1, 3, 7, 21, 29, 87, 203, 609
  • Factor pairs: 609 = 1 x 609, 3 x 203, 7 x 87, or 21 x 29
  • 609 has no square factors that allow its square root to be simplified. √609 ≈ 24.677925.

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609 Factors

608 and Level 1

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608 is the sum of 19 consecutive numbers:

23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 +33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 = 608. What does that have to do with the factors of 608?

608 Puzzle

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

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  • 608 is a composite number.
  • Prime factorization: 608 = 2 x 2 x 2 x 2 x 2 x 19, which can be written 608 = (2^5) x 19
  • 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 608 has exactly 12 factors.
  • Factors of 608: 1, 2, 4, 8, 16, 19, 32, 38, 76, 152, 304, 608
  • Factor pairs: 608 = 1 x 608, 2 x 304, 4 x 152, 8 x 76, 16 x 38, or 19 x 32
  • Taking the factor pair with the largest square number factor, we get √608 = (√16)(√38) = 4√38 ≈ 24.657656.

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608 Factors

607 and Level 6

/ ivasallay / Leave a comment

607 is the sum of the three prime numbers from 197 to 211.

607 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 607 is a prime number.
  • Prime factorization: 607 is prime.
  • The exponent of prime number 607 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 607 has exactly 2 factors.
  • Factors of 607: 1, 607
  • Factor pairs: 607 = 1 x 607
  • 607 has no square factors that allow its square root to be simplified. √607 ≈ 24.637

How do we know that 607 is a prime number? If 607 were not a prime number, then it would be divisible by at least one prime number less than or equal to √607 ≈ 24.6. Since 607 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 607 is a prime number.

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607 Logic

606 and Level 5

/ ivasallay / Leave a comment

606 is the sum of the six prime numbers from 89 to 109

606 is also the hypotenuse of the Pythagorean triple 120-594-606. What is the greatest common factor of those three numbers?

606 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 606 is a composite number.
  • Prime factorization: 606 = 2 x 3 x 101
  • 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 606 has exactly 8 factors.
  • Factors of 606: 1, 2, 3, 6, 101, 202, 303, 606
  • Factor pairs: 606 = 1 x 606, 2 x 303, 3 x 202, or 6 x 101
  • 606 has no square factors that allow its square root to be simplified. √606 ≈ 24.617067

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605 and Level 4

/ ivasallay / Leave a comment

605 is the hypotenuse of the Pythagorean triple 363-484-605. The greatest common factor of those three numbers should be fairly easy to spot.

6 – 0 + 5 = 11, obviously a multiple of 11, so 605 passes 11’s divisibility test and can be evenly divided by 11.

Coincidentally, OEIS.org shared some trivia about 605’s digits: 6 + 0 + 5 = 11, the greatest prime factor of 605.

605 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 605 is a composite number.
  • Prime factorization: 605 = 5 x 11 x 11, which can be written 605 = 5 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 605 has exactly 6 factors.
  • Factors of 605: 1, 5, 11, 55, 121, 605
  • Factor pairs: 605 = 1 x 605, 5 x 121, or 11 x 55
  • Taking the factor pair with the largest square number factor, we get √605 = (√121)(√5) = 11√5 ≈ 24.5967

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605 Logic

604 and Level 3

/ ivasallay / Leave a comment

604 is not a power of 2, can be evenly divided by 4, but not by 8, and 604 is greater than 28. Those three facts together mean that 604 can be written as the sum of 8 consecutive whole numbers. To find the 8 numbers first divide 604 by 8 to get 75 1/2, then add the 4 counting numbers before 75 1/2 to the 4 counting numbers after it:

72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 = 604

604 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 604 is a composite number.
  • Prime factorization: 604 = 2 x 2 x 151, which can be written 604 = (2^2) x 151
  • 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 604 has exactly 6 factors.
  • Factors of 604: 1, 2, 4, 151, 302, 604
  • Factor pairs: 604 = 1 x 604, 2 x 302, or 4 x 151
  • Taking the factor pair with the largest square number factor, we get √604 = (√4)(√151) = 2√151 ≈ 24.576411

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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.

604 Factors

603 Some Consecutive Numbers and Level 2

/ ivasallay / Leave a comment

603 has six factors. So do the next two counting numbers. Many numbers have exactly six factors, but something unique happens this time.

The most common time a number will have six factors is when that number is the product of a squared prime number and a different prime number. That happens a lot, but this is the first time it has happened to three consecutive numbers!

603, 604, 605 Consecutive

It is NOT the first time that three or even four consecutive numbers had 6 factors because there is also another, less common way for a number to do that: when a number is equal to a prime number raised to the 5th power. Look at this list of consecutive numbers with 6 factors.

242, 243, 244, 245

Between 245 and 603 there are not any other lists of three or more consecutive numbers with six factors.

603 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 603 is a composite number.
  • Prime factorization: 603 = 3 x 3 x 67, which can be written 603 = (3^2) x 67
  • 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 603 has exactly 6 factors.
  • Factors of 603: 1, 3, 9, 67, 201, 603
  • Factor pairs: 603 = 1 x 603, 3 x 201, or 9 x 67
  • Taking the factor pair with the largest square number factor, we get √603 = (√9)(√67) = 3√67 ≈ 24.556058

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603 Factors