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