A Multiplication Based Logic Puzzle

9 – 6 + 8 = 11, so 968 is divisible by 11. In fact, 11 is its largest prime factor so we can make a beautiful factor cake for 968 with two candles on top.

Guess what, 968 is also divisible by 11², so its factor cake can have even more candles!

Look at its factor cake. Notice that each of 968’s prime factors is repeated at least once. 2³ × 11² = 968

Stetson.edu alerts us to the fact that 968 is the twelfth Achilles number. That means that each of its prime factors has an exponent greater than one yet the greatest common factor of those exponents is still one. (Perfect squares, cubes, etc. are not Achilles numbers.)

There are thirteen Achilles numbers less than 1000. Here is a chart of them and their prime factorizations. These numbers appear to be few and far between. The previous Achilles number was 104 less than 968, but the next one is only 4 numbers away!

Being only 4 numbers away is pretty amazing. Consecutive Achilles numbers actually exist. You can find the smallest pair of them in the Wikipedia article. Both numbers are greater than 5 billion. Again, being only 4 numbers away is pretty amazing.

The smallest Achilles number made with three different prime numbers raised to various powers is 2³·3²·5² = 1800. Notice that each exponent is greater than one yet the greatest common factor of those exponents is still one.

Here’s a little more about the number 968:

I like the way 968 looks in a few other bases:
It’s 2552 in BASE 7 because 2(7³) + 5(7²) + 5(7¹) + 2(7⁰) = 968,
800 in BASE 11 because 8(11²) = 8(121) = 968,
242 in BASE 21 because 2(21²) + 4(21¹) + 2(21⁰) = 968,
200 in BASE 22 because 2(22²) = 2(484) = 968.

  • 968 is a composite number.
  • Prime factorization: 968 = 2 × 2 × 2 × 11 × 11, which can be written 968 = 2³ × 11²
  • The exponents in the prime factorization are 3 and 2. Adding one to each and multiplying we get (3 + 1)(2 + 1) = 4 × 3 = 12. Therefore 968 has exactly 12 factors.
  • Factors of 968: 1, 2, 4, 8, 11, 22, 44, 88, 121, 242, 484, 968
  • Factor pairs: 968 = 1 × 968, 2 × 484, 4 × 242, 8 × 121, 11 × 88, or 22 × 44
  • Taking the factor pair with the largest square number factor, we get √968 = (√484)(√2) = 22√2 ≈ 31.11269837

 

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