Level 2 Puzzle

623 and Level 2

/ ivasallay / 2 Comments

623 is the hypotenuse of Pythagorean triple 273-560-623. What is the greatest common factor of those three numbers?

623 is not divisible by 2, 3, or 5. Is 623 divisible by 7? You can apply either of the following divisibility rules after you separate 623’s digits into 62 and 3:

1st rule: 62 – 2(3) = 62 – 6 = 56. Since 56 is divisible by 7, 623 is divisible by 7.

2nd rule: 62 + 5(3) = 62 + 15 = 77. Since 77 is divisible by 7, 623 is divisible by 7.

86 + 87 + 88 + 89 + 90 + 91 + 92 = 623  (7 consecutive numbers because 623 is divisible by 7)

623 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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  • 623 is a composite number.
  • Prime factorization: 623 = 7 x 89
  • 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 623 has exactly 4 factors.
  • Factors of 623: 1, 7, 89, 623
  • Factor pairs: 623 = 1 x 623 or 7 x 89
  • 623 has no square factors that allow its square root to be simplified. √623 ≈ 24.9599679.

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623 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

 

609 and Level 2

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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

603 Some Consecutive Numbers and Level 2

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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

594 and Level 2

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It is as easy as 1-2-3 to remember this number fact from OEIS.org:

(1^5) + (2^9) + (3^4) = 594

594 is the sum of the ten prime numbers from 41 to 79.

594 is even so it is divisible by 2.

5 + 9 + 4 = 18 so 594 is divisible by both 3 and 9 (and by 6 and 18 because 594 is even)

5 – 9 + 4 = 0 so 594 is divisible by 11.

594 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 594 is a composite number.
  • Prime factorization: 594 = 2 x 3 x 3 x 3 x 11, which can be written 594 = 2 x (3^3) x 11
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 594 has exactly 16 factors.
  • Factors of 594: 1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 99, 198, 297, 594
  • Factor pairs: 594 = 1 x 594, 2 x 297, 3 x 198, 6 x 99, 9 x 66, 11 x 54, 18 x 33, or 22 x 27
  • Taking the factor pair with the largest square number factor, we get √594 = (√9)(√66) = 3√66 ≈ 24.372115

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594 Factors

587 and Level 2

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587 is the sum of the five prime numbers from 107 to 131.

587 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 587 is a prime number.
  • Prime factorization: 587 is prime.
  • The exponent of prime number 587 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 587 has exactly 2 factors.
  • Factors of 587: 1, 587
  • Factor pairs: 587 = 1 x 587
  • 587 has no square factors that allow its square root to be simplified. √587 ≈ 24.22808

How do we know that 587 is a prime number? If 587 were not a prime number, then it would be divisible by at least one prime number less than or equal to √587 ≈ 24.2. Since 587 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 587 is a prime number.

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587 Factors

580 and Level 2

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580 is the sum of all the prime numbers from 83 to 107. Do you know what those consecutive prime numbers are?

580 is also the hypotenuse of four Pythagorean triples. In each case can you find the factor of 580 that is the greatest common factor of the triple?

  • 68-576-580
  • 96-572-580
  • 348-464-580
  • 400-420-580

580 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-10

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  • 580 is a composite number.
  • Prime factorization: 580 = 2 x 2 x 5 x 29, which can be written 580 = (2^2) x 5 x 29
  • 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 580 has exactly 12 factors.
  • Factors of 580: 1, 2, 4, 5, 10, 20, 29, 58, 116, 145, 290, 580
  • Factor pairs: 580 = 1 x 580, 2 x 290, 4 x 145, 5 x 116, 10 x 58, or 20 x 29
  • Taking the factor pair with the largest square number factor, we get √580 = (√4)(√145) = 2√145 ≈ 24.083189

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580 Factors

573 and Level 2

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573 is made from three consecutive odd numbers so it is divisible by 3. The number in the middle, 5, is not divisible by 3, so 573 is not divisible by 9.

573 squared is 328,329, a 6 digit number that looks like two consecutive numbers! Thank you OEIS.org for that interesting fact.

573 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-03

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  • 573 is a composite number.
  • Prime factorization: 573 = 3 x 191
  • 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 573 has exactly 4 factors.
  • Factors of 573: 1, 3, 191, 573
  • Factor pairs: 573 = 1 x 573 or 3 x 191
  • 573 has no square factors that allow its square root to be simplified. √573 ≈ 23.9374

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573 Factors

566 and Level 2

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566 is the sum of all the prime numbers from 3 to 67.

566 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 566 is a composite number.
  • Prime factorization: 566 = 2 x 283
  • 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 566 has exactly 4 factors.
  • Factors of 566: 1, 2, 283, 566
  • Factor pairs: 566 = 1 x 566 or 2 x 283
  • 566 has no square factors that allow its square root to be simplified. √566 ≈ 23.79075

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566 Factors

 

559 and Level 2

/ ivasallay / 2 Comments

All the prime numbers from 67 to 97 add up to 559, and all the prime numbers from 103 to 127 also add up to 559.

559 is also the hypotenuse of Pythagorean triple 215-516-559. What is the greatest common factor of those three numbers?

559 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 559 is a composite number.
  • Prime factorization: 559 = 13 x 43
  • 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 559 has exactly 4 factors.
  • Factors of 559: 1, 13, 43, 559
  • Factor pairs: 559 = 1 x 559 or 13 x 43
  • 559 has no square factors that allow its square root to be simplified. √559 ≈ 23.64318

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559 Factors