Multiplication table

604 and Level 3

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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

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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

602 and Level 1

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(5^602) is a very large number with 421 digits. What are the first three of those 421 digits? 602. Thank you, OEIS.org for that interesting number fact.

602 Puzzle

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

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  • 602 is a composite number.
  • Prime factorization: 602 = 2 x 7 x 43
  • 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 602 has exactly 8 factors.
  • Factors of 602: 1, 2, 7, 14, 43, 86, 301, 602
  • Factor pairs: 602 = 1 x 602, 2 x 301, 7 x 86, or 14 x 43
  • 602 has no square factors that allow its square root to be simplified. √602 ≈ 24.535688.

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602 Factors

 

598 and Level 6

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598 is the sum of all the prime number from 17 to 71.

598 is also the hypotenuse of the Pythagorean triple 230-552-598. What is the greatest common factor of those three numbers?

Here’s another easy as 1-2-3 number fact from OEIS.org: (5^1) + (9^2) +(8^3) = 598.

Since 23 + 3 = 26 and 23 x 26 = 598, we know we are only 2 away from 24 x 25 = 600.

598 Puzzle

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

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  • 598 is a composite number.
  • Prime factorization: 598 = 2 x 13 x 23
  • 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 598 has exactly 8 factors.
  • Factors of 598: 1, 2, 13, 23, 26, 46, 299, 598
  • Factor pairs: 598 = 1 x 598, 2 x 299, 13 x 46, or 23 x 26
  • 598 has no square factors that allow its square root to be simplified. √598 ≈ 24.4540385.

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598 Logic

597 and Level 5

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597 is made from three consecutive odd numbers so it is divisible by 3.

Because it is divisible by 3, it is the sum of three consecutive numbers. It is also the sum of three consecutive odd numbers:

  • 198 + 199 + 200 = 597
  • 197 + 199 + 201 = 597

Why was 199 the middle number in both of those sums?

OEIS.org informs us that even though (597!!! + 1) generates an error message on our calculators, it is indeed a very large prime number. Whoever discovered that fact must have been extremely excited.

597 Puzzle

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

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  • 597 is a composite number.
  • Prime factorization: 597 = 3 x 199
  • 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 597 has exactly 4 factors.
  • Factors of 597: 1, 3, 199, 597
  • Factor pairs: 597 = 1 x 597 or 3 x 199
  • 597 has no square factors that allow its square root to be simplified. √597 ≈ 24.43358

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597 Logic

596 Select the Correct Top Row

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596 is the sum of the four primes numbers from 139 to 157. The two prime numbers in the middle of that sum are twin primes. Do you know what they are?

596 is also the hypotenuse of the Pythagorean triple 204-560-596. What is the greatest common factor of those three numbers?

596 Puzzle

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

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  • 596 is a composite number.
  • Prime factorization: 596 = 2 x 2 x 149, which can be written 596 = (2^2) x 149
  • 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 596 has exactly 6 factors.
  • Factors of 596: 1, 2, 4, 149, 298, 596
  • Factor pairs: 596 = 1 x 596, 2 x 298, or 4 x 149
  • Taking the factor pair with the largest square number factor, we get √596 = (√4)(√149) = 2√149 ≈ 24.41311123

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596 Logic

595 and Level 3

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595 = (34/2)(34 + 1) so 595 is the 34th triangular number.

Thus 1 + 2 + 3 + 4 + . . . . . + 31 + 32 + 33 + 34 = 595

595 is also the hypotenuse of four Pythagorean triples. Each of these triples has its own greatest common factor.

  • 91-588-595
  • 252-539-595
  • 280-525-595
  • 357-476-595

595 Puzzle

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

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  • 595 is a composite number.
  • Prime factorization: 595 = 5 x 7 x 17
  • 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 595 has exactly 8 factors.
  • Factors of 595: 1, 5, 7, 17, 35, 85, 119, 595
  • Factor pairs: 595 = 1 x 595, 5 x 119, 7 x 85, or 17 x 35
  • 595 has no square factors that allow its square root to be simplified. √595 ≈ 24.39262.

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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.

595 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

593 and Level 1

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593 is the sum of the seven prime numbers from 71 to 101 as well as the nine prime numbers from 47 to 83.

When the number formed from the last two digits of prime number 593 is divided by 4, the remainder is 1 so 593 can be expressed at the sum of two squares AND is the hypotenuse of a primitive Pythagorean triple. Specifically:

23² + 8² = 593

593 is the hypotenuse of the primitive Pythagorean triple prime 368-465-593

593 Puzzle

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

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  • 593 is a prime number.
  • Prime factorization: 593 is prime.
  • The exponent of prime number 593 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 593 has exactly 2 factors.
  • Factors of 593: 1, 593
  • Factor pairs: 593 = 1 x 593
  • 593 has no square factors that allow its square root to be simplified. √593 ≈ 24.35159

How do we know that 593 is a prime number? If 593 were not a prime number, then it would be divisible by at least one prime number less than or equal to √593 ≈ 24.4. Since 593 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 593 is a prime number.

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593 Factors

590 and Level 5

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3(20^2)/2 – 20/2 = 600 – 10 = 590. That means that 590 is the 20th pentagonal number.

590 is also the hypotenuse of the Pythagorean triple 354-472-590. Which factor of 590 is the greatest common factor of those three numbers? Hint: In this case the GCF is the difference between the middle number in the triple and either of the other two numbers.

590 Puzzle

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

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  • 590 is a composite number.
  • Prime factorization: 590 = 2 x 5 x 59
  • 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 590 has exactly 8 factors.
  • Factors of 590: 1, 2, 5, 10, 59, 118, 295, 590
  • Factor pairs: 590 = 1 x 590, 2 x 295, 5 x 118, or 10 x 59
  • 590 has no square factors that allow its square root to be simplified. √590 ≈ 24.2899

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590 Logic