278 MR. G-. T. BEIfNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Example .—Residues of powers of (1 + ^) for mod 100 = (1 + ?)^ (2 + (1 + 2{)-, 
= (1 + if, 
and therefore 
r = 4, 
P = (2 + ( 1+ 2^)^ 
therefore 
Similarly 
therefore 
and therefore 
1 i has exp 4 mod 2 + { and (1 if [mod (2 + 1)^], 
1 + has exp 20 mod (2 + if. 
1 + { has exp 4 mod 1 + 2^ and (1 if ^ 1 [mod (1 4* 2i)~], 
1 + ?’ has exp 20 mod (1 + 2if, 
t = 20. 
The residues are 
1+7 27 98 + 27 96 96 + 967 927 8 + 927 16 16 + 167 327 
68 + 327 36 36 + 367 727 28 + 727 56 56 + 567 127 
88 + 127 76 76 + 767 527 48 + 527. 
Example. —The residues of the powders of 688 + 7847 for mod 1000. 
1000 = - (1 + if (2 + if (1 + 27)3. 
688 + 7847 = 0 [mod (1 + if]. 
688 + 7847= 688 — 57 (784) [mod (2 + 7)®]. (See preface.) 
= 0 [mod (2 + 7)3]. 
d 
688 + 7847 = 688 + 57 (784) [mod (1 + 27)3]. 
= 1 [mod (1 + 27)3]. 
Hence, r = 1 and 7=1, and so all powers of the number 688 + 7847 are congruent 
to itself for mod 1000. 
688 
688 
784 
688 
784 
784 
504 
752 
136 
04 
04 
72 
8 
6 
8 
344 
392 
656 
2 
784 
