A Multiplication Based Logic Puzzle

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

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