#math trivia #125 solution

The day-numbers that divide a power of 10 (and the power they divide) are:

  • 1 — 1
  • 2 — 10
  • 4 — 100
  • 5 — 10
  • 8 — 1000
  • 10 — 10
  • 16 — 10,000
  • 20 — 100
  • 25 — 100
  • 32 — 100,000
  • 40 — 1000
  • 50 — 100
  • 64 — 1,000,000
  • 80 — 10,000
  • 100 — 100
  • 125 — 1000
  • 128 — 10,000,000
  • 160 — 100,000
  • 200 — 1000
  • 250 — 1000
  • 256 — 100,000,000
  • 320 — 1,000,000

The general pattern is a day-number divides a power of 10 if and only if it has the form 2^x * 5^y for x, y >= 0.  The smallest power of 10 that the day-number divides is 10^z where z is the larger of x and y.  If the day-number is divisible by any prime numbers other than 2 and 5, it cannot divide a power of 10, because powers of 10, being of the form 10^z = 2^z * 5^z, are only divisible by powers of 2, powers of 5, and their products.

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