Congruences Modulo help (1 Viewer)

[maokao]

Could someone please help me with this question:

Find the remainder that the number (59^59) + (87^87) - (115^115) leaves when it is divided by 29.

Thanks

 

[Sy123]

I'm not very experienced with modulo and congruency, but using the 4 properties that I do know:









I've named the properties respectively:

First, we notice by inspection that:



And since by (3), we can multiply the congruencies side by side, simply multiply them side by side 59 times:



Now, we also notice that:



Multiply 87 of these side by side (by property (3))



Now, notice that:



Multiply side by side by property (3)



Now my property (1)



So we now have 3 congruencies:







Add them side by side by property (2):



Translate this into the fact that:



Says that (a-b)/m is an integer, so:

(*)

for integer k.

We are looking for some r integer where of the form:



Rearranging (*), we find that the remainder is 2.

(I hope this is right)

 

[maokao]

Thanks

 

[tywebb]

Yeah. It's 2.

((59^59) + (87^87) - (115^115)-2)/29
=-32 949 865 242 422 507 823 452 269 476 761 313 150 573 966 290 295 782 281 360 311 616 802 532 601 819 644 444 159 331 635 763 419 680 913 921 897 320 261 499 521 768 973 227 527 990 336 727 893 870 031 676 014 033 403 704 318 608 592 423 362 036 475 427 826 637 023 672 892 492 097 248 777 053 417 035 018 871 295

which is a whole number.