#math trivia #134 solution

#math trivia for #May13:  #134 and its prime divisors (67, 2) have no common digits.  When did this last happen?  When will it happen again?

Answer:  This is a fun problem that can be solved with by trial-and-error moving forward and backward from 134, using some properties of factorizations to shorten the search.

Clearly, a prime can’t be a solution, because it’s its own prime divisor, so it has the same digits.  A non-prime ending in 5 won’t work either, nor will one ending in 2, nor any even number with a 2 in it. Also, there can be at most one of each prime divisor, so powers won’t work, nor will numbers divisible by powers.

So the near values less than 134 to test include:

  • 133:  7*19 — no
  • 130:  2*5*13 — no
  • 123:  3*41 — no
  • 119:  7*17 — no
  • 118:  2*59 — yes — this is the last one

The ones greater than 134 include:

  • 138:  2*3*23 — no (this one is triply repetitive — could be a good choice for another problem!)
  • 143:  11*13 — no (another triple)
  • 146:  2*73 — yes — this is the next one

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