cinelli wrote:If you evaluate 19^92 and write it in the usual decimal form, what are the final six digits?
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The final 6 digits of a number x are just x MOD 1000000 (i.e. the remainder when it's divided by 1000000), with leading zeros added if necessary, and the MOD function has the property that (xy) MOD N = ((x MOD N) * (y MOD N)) MOD N.
That gives us in succession, starting with 19 MOD 1000000 = 19 and using a calculator:
19^2 MOD 1000000 = 19*19 MOD 1000000 = 361
19^3 MOD 1000000 = 19*361 MOD 1000000 = 6859
19^5 MOD 1000000 = 361*6859 MOD 1000000 = 2476099 MOD 1000000 = 476099
19^10 MOD 1000000 = 476099*476099 MOD 1000000 = 226670257801 MOD 1000000 = 257801
19^20 MOD 1000000 = 257801*257801 MOD 1000000 = 66461355601 MOD 10000 = 355601
19^23 MOD 1000000 = 355601*6859 MOD 1000000 = 2439067259 MOD 1000000 = 67259
19^46 MOD 1000000 = 67259*67259 MOD 1000000 = 4523773081 MOD 1000000 = 773081
19^92 MOD 1000000 = 773081*773081 MOD 1000000 = 597654232561 MOD 10000 = 232561
So the answer is that the last six digits are 232561.
Gengulphus