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 Mod And Near Myriad (Posted on 2011-04-24)
Determine all possible positive integer(s) N < 10,000, such that:

2N = 88 (mod 167), and:

2N = 70 (mod 83)

**** For an extra challenge, solve this problem without using a computer program.

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution using Frink | Comment 2 of 4 |

lhs=1; prev=0
for n= 1 to 99999
{
lhs=lhs*2
if lhs mod 167 == 88 and lhs mod 83 ==70
{
l=n*log[2]
diff=n-prev
prev=n
println ["\$n \$l \$diff"]
}
}

finds only two that are below 10,000: 2004 and 8810, leading to a 604-digit number and to a 2653-digit number as their respective powers of 2.

Those two values of n are separated by 6806, and apparently every 6806th value of n thereafter satisfies the conditions as given in this table:

`  n       log(2^n)             dn 2004   603.26411131061822422 2004 8810  2652.0742617996739298  680615616  4700.8844122887296354  680622422  6749.694562777785341   680629228  8798.5047132668410466  680636034 10847.314863755896752   680642840 12896.125014244952458   680649646 14944.935164734008163   680656452 16993.745315223063869   680663258 19042.555465712119575   680670064 21091.36561620117528    680676870 23140.175766690230986   680683676 25188.985917179286692   680690482 27237.796067668342397   680697288 29286.606218157398103   6806  `

the ceiling of the common log can be used to show the number of digits in the power of 2, and the fractional part allowing view of the beginning of the number by taking the antilog.

 Posted by Charlie on 2011-04-24 14:28:41

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