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 Probable Ten Power (Posted on 2009-10-06)
Determine the probability that for a positive base ten integer X drawn at random between 1 and 100 inclusively, the number 10X is expressible as the product of precisely two positive integers, neither of which contains the digit zero.

 No Solution Yet Submitted by K Sengupta Rating: 4.0000 (1 votes)

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 computer solution | Comment 1 of 2

The probability is 1/10 as there are 10 values of x that work: 1, 2, 3, 4, 5, 6, 7, 9, 18 and 33.

The program logic depends on the fact that neither factor chosen can have both 2 and 5 as factor within itself, and so they must be 2^x and 5^x.

`list   10   for X=1 to 100   15    Y=10^X   20    Dvr=2^X   30    Q=5^X   40    S=cutspc(str(Dvr)+str(Q))   50    if instr(S,"0")=0 then   60           :print X;Y;Dvr;Q   70           :Ct=Ct+1   90   next X  100   print CtOKrun x  10^x  factors 1  10  2  5 2  100  4  25 3  1000  8  125 4  10000  16  625 5  100000  32  3125 6  1000000  64  15625 7  10000000  128  78125 9  1000000000  512  1953125 18  1000000000000000000  262144  3814697265625 33  1000000000000000000000000000000000  8589934592  116415321826934814453125`
` 10 (count of values)`

Increasing the limit beyond x=100, to the limit of UBASIC, shows no other possible values of x through 1549, so one can hypothesize that there are no more such values ever, but that's just that--a hypothesis:

list
10   for X=1 to 2100
15    Y=10^X
20    Dvr=2^X
30    Q=5^X
40    S1=cutspc(str(Dvr)):S2=str(Q)
50    if instr(S1,"0")=0 and instr(S2,"0")=0 then
60           :print X;Y;Dvr;Q
70           :Ct=Ct+1
90   next X
100   print Ct
OK
run
1  10  2  5
2  100  4  25
3  1000  8  125
4  10000  16  625
5  100000  32  3125
6  1000000  64  15625
7  10000000  128  78125
9  1000000000  512  1953125
18  1000000000000000000  262144  3814697265625
33  1000000000000000000000000000000000  8589934592  116415321826934814453125
Overflow in 40
?x
1550

 Posted by Charlie on 2009-10-06 12:17:28

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