Modulo - rekning

[Mattebruker]

Rekn ut 10^42 mod 61

Hint: 42 = 2 + 8 + 32

[MaximilliaVeum]

Hello Mattebruker. Your question is quite complex. I hope this helps:
61=308−(61×5)=308−305=3
308mod61=308−(61×5)=308−305=3
....

[Mattebruker]

Hello !

I don't realize where the number ( 308 ) comes from. You should rather take into account the hint given in text ( 42 = 2 + 8 + 32 ).
Maybe this rewriting will lead you onto a more fruitful trace:

10^42 = 10^2 * 10^8 * 10^32

Another hint: 10^2 = 100 " KONGRUENT " 39 ( MOD 61 )

10^4 = ( 10^2 )^ 2 " KONGRUENT " 39^2 = ( 40 - 1 )^2 = 40^2 - 2 * 40 * 1 + 1 = 1521 = 1525 - 4 = ( 61 * 25 - 4) " kongruent " -4 ( mod 61 )

Do you see how to move forward ? Good luck !

[salimnl]

Hello !
Incredibox
I don't realize where the number ( 308 ) comes from. You should rather take into account the hint given in text ( 42 = 2 + 8 + 32 ).
Maybe this rewriting will lead you onto a more fruitful trace:

10^42 = 10^2 * 10^8 * 10^32

Another hint: 10^2 = 100 " KONGRUENT " 39 ( MOD 61 )

10^4 = ( 10^2 )^ 2 " KONGRUENT " 39^2 = ( 40 - 1 )^2 = 40^2 - 2 * 40 * 1 + 1 = 1521 = 1525 - 4 = ( 61 * 25 - 4) " kongruent " -4 ( mod 61 )

Do you see how to move forward ? Good luck !

I have a small question, could you explain more about how "kongruent" is applied in this problem? It feels like there is a special connection between the operations and I would like to understand this method more deeply.