Digits 251 in sequence (Posted on 2009-01-05)

	Submitted by pcbouhid
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If n * 0. ...251... = integer, then n * 0.251... = integer (where we simply delete the digits before 251). So it is sufficient to consider 0.251... If n * 0.251 = integer, then n must be a multiple of 1000. So assume n ≤ 1000. We require that [n * 0.251] = m, [n * 0.252] = m+1, for some m. Let {x} denote the fractional part of x. Then since n * 0.252 - n * 0.251 = n/1000, we must have {n * 0.251} ≥ 1 - n/1000. If n is a multiple of 4, then {n * 0.251} = {n/1000}, so we need n > 500. If n = 1 mod 4, then {n * 0.251} = {0.25 + n/1000}, so n > 375. Similarly, if n = 2 mod 4, then n > 250, and if n = 3 mod 4, then n > 125. Thus the smallest candidate is 127, and 32/127 = 0.251...

	Subject	Author	Date
	Puzzle Answer	K Sengupta	2022-07-28 04:26:54
	re(2): upper bound	brianjn	2009-01-07 05:41:50
	non-computer solution?	Gamer	2009-01-06 04:01:28
	computer solution	Charlie	2009-01-05 15:52:08
	re: upper bound	Sing4TheDay	2009-01-05 15:42:54
	upper bound	Paul	2009-01-05 15:10:46