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        riddles >> medium >> Same Leading Digit
        
(Message started by: THUDandBLUNDER on Sep 5th, 2004, 7:04am)
Title: Same Leading Digit
Post by THUDandBLUNDER on Sep 5th, 2004, 7:04am
When n = 0
2n = 1
5n = 1
(Same leading digit)

When n = 5
2n = 32
5n = 3125
(Same leading digit)

Can you find any more such examples?
Title: Re: Same Leading Digit
Post by Barukh on Sep 5th, 2004, 9:30am
[hide]n=15[/hide].
Title: Re: Same Leading Digit
Post by towr on Sep 5th, 2004, 12:41pm
::[hide]
searching up to n = 200

n, 2n, 5n

0, 1, 1

5, 32, 3125

15, 32768, 30517578125

78, 302231454903657293676544, 3308722450212110699485634768279851414263248443603515625

88, 309485009821345068724781056, 32311742677852643549664402033982923967414535582065582275390625

98, 316912650057057350374175801344, 315544362088404722164691426113114491869282574043609201908111572265625

108, 324518553658426726783156020576256, 3081487911019577364889564708135883709660962637144621112383902072906494140625

118, 332306998946228968225951765070086144, 30092655381050560203999655352889489352157838253365440550624043680727481842041015625

181, 3064991081731777716716694054300618367237478244367204352, 3262652233999262263355147054628273777850582129034483273879318227734861622298743113611448063426934140807134099304676055908203125

191, 3138550867693340381917894711603833208051177722232017256448, 31861838222649045540577607955354236111822091103852375721477717067723258030261163218861797494403653718819668938522227108478546142578125
[/hide]::
Title: Re: Same Leading Digit
Post by Barukh on Sep 6th, 2004, 4:25am
It is not difficult to see that there are infinitely many such numbers.

Using [hide]continued fractions[/hide] one can find fairly big examples without actual search (but of course not all such numbers!).

Here're a few: [hide]14369, 56408, 127185, 3053508, 22349997[/hide].


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