A Multiplication Based Logic Puzzle

Posts tagged ‘centered square number’

761 and Level 1

  • 761 is a prime number.
  • Prime factorization: 761 is prime.
  • The exponent of prime number 761 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 761 has exactly 2 factors.
  • Factors of 761: 1, 761
  • Factor pairs: 761 = 1 x 761
  • 761 has no square factors that allow its square root to be simplified. √761 ≈ 27.586228.

761-factor-pairs

How do we know that 761 is a prime number? If 761 were not a prime number, then it would be divisible by at least one prime number less than or equal to √761 ≈ 27.6. Since 761 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 761 is a prime number.

There will be a lot about squares in today’s post, and all of the clues in today’s puzzle are perfect squares, too:

761 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s more about the number 761:

Because 761 is a prime number, there is only one way to write it as the sum of consecutive numbers: 380 + 381 = 761.

However, because it is a centered square number, 761 can be written as the sum of consecutive square numbers: 20² + 19² = 761.

761 Centered Square Number

And because it is the sum of two consecutive square numbers, it is also the hypotenuse of a primitive Pythagorean triple: 39-760-761 which was calculated using 20² – 19², 2(20)(19), 20² + 19².

Notice also that 20 + 19 = 39, ⌊39²/2⌋ = 760, and ⌈39²/2⌉ = 761. (Those peculiar parenthesis mean⌊round down to the closest whole number⌋ and ⌈round up to the closest whole number⌉.)

In the 39 x 39 graphic shown above there are 760 white squares and 761 squares with some color. Thus, looking at that graphic is one way to look at the primitive Pythagorean triple 39-760-761.

All of that gives us another way to know that 761 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 20² + 19² = 761 with 20 and 19 having no common prime factors, 761 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √761 ≈ 27.6. Since 761 is not divisible by 5, 13, or 17, we know that 761 is a prime number.

761 can also be expressed as the sum of three squares TEN ways:

  • 27² + 4² + 4² = 761
  • 26² + 9² + 2² = 761
  • 26² + 7² + 6² = 761
  • 25² + 10² + 6² = 761
  • 24² + 13² + 4² = 761
  • 24² + 11² + 8² = 761
  • 23² + 14² + 6² = 761
  • 22² + 14² + 9² = 761
  • 21² + 16² + 8² = 761
  • 19² + 16² + 12² = 761

761 is a palindrome in two different bases:

  • 535 BASE 12; note that 5(144) + 3(12) + 5(1) = 761.
  • 1I1 BASE 20 (I = 18 base 10); note that 1(400) + 18(20) + 1(1) = 761.

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761 Factors

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685 Is the Sum of Two Squares Two Different Ways

Because both of its prime factors are hypotenuses of primitive Pythagorean triples, 685 is the hypotenuse of FOUR Pythagorean triples. Two are primitive; two are not:

  • 37-684-685 which was calculated from 19² – 18², 2(19)(18), 19² + 18²
  • 156-667-685 which was calculated from 2(26)(3), 26² – 3², 26² + 3²
  • 411-548-685 (What factor of 685 is the greatest common factor of those 3 numbers?)
  • 440-525-685 (and what is their greatest common factor?)

As you may have notice from those calculations, 685 is the sum of two squares two different ways:

  • 19² + 18² = 685
  • 26² + 3² = 685

685 is the 19th centered square number because 18 and 19 are consecutive numbers and 19² + 18² = 685. There are 685 small squares of various colors in this graphic.

685 is the 19th Centered Squared Number

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  • 685 is a composite number.
  • Prime factorization: 685 = 5 x 137
  • 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 685 has exactly 4 factors.
  • Factors of 685: 1, 5, 137, 685
  • Factor pairs: 685 = 1 x 685 or 5 x 137
  • 685 has no square factors that allow its square root to be simplified. √685 ≈ 26.172505.

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