R is a rational number such that 2^123456789 degrees Celsius is precisely equal to R degrees Fahrenheit.

(I) Determine the first two digits (reading left to right) in [R].

(II) Determine the last two digits (reading left to right) in [R].

(III) Determine the first digit (reading left to right) following the decimal point in R.

Note: [x] denotes the greatest integer ≤ x.

print modpow(2,123456789,10000000000000000)
 8355204473946112

determines that the last few digits of 2^123456789 are  ...8355204473946112.

When divided by 5, only the first of these digits will be affected depending on the digits before.

So, when divided by 5, the number ends

...71040894789222.4

and then when multiplied by 9,

... 639368053103001.6

Then you add 32, making the ending ... 639368053103033.6

So part (II): 33

and Part (III): 6

More analytic solution:

2^n mod 1000 has a cycle of length 100 starting with 2^3:

 8  16  32  64  128  256  512  24  48  96  192  384  768  536  72  144  288  576  152  304  608  216  432  864  728  456  912  824  648  296  592  184  368  736  472  944  888  776  552  104  208  416  832  664  328  656  312  624  248  496  992  984  968  936  872  744  488  976  952  904  808  616  232  464  928  856  712  424  848  696  392  784  568  136  272  544  88  176  352 704  408  816  632  264  528  56  112  224  448  896  792  584  168  336  672 344  688  376  752  504

so 2^123456789 mod 1000 is the same if taken mod 89, or for that matter, 189, which is 3*3*3*7, so you could cube, cube, cube and raise to the 7th. In any event, the power, mod 1000 is 112, so the last two digits after dividing by 5 are 22, with 2 left over, so the ending is ...22.4. Then multiplication by 9 gives ...01.6, with the addition of 32 then giving the same 33 and 6 as the answers to parts (II) and (III).

Posted by Charlie on 2010-12-18 17:15:35