#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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