Use the fact that 29 16 8 4 1 to compute 1829 mod 65Solu

Use the fact that 29 = 16 + 8 + 4 + 1 to compute 18²⁹ mod 65.

65= 4*16+1

29=16+8+4+1

4*29= 4*(16+8+4+1) = 65+ 4*(8+4)+3*1=65+51

1829 = 15*(4*29)+89

So 1829 modulo 65 = (15*(65+51)+89) modulo 65

(15*65+15*51+89) modulo 65 = (15*51+24) modulo 65 = 789 modulo 65 = 9