A 552 Antoine Verroken Q1. représentation en binaire de 105 est 1101001 105 = 1 + 8 + 32 + 64
a. divisible par 7
1. 3^8 = 3² * 3² * 3² * 3² = 2 * 2 * 2 * 2 = 2 ( mod 7 ) 3^32 = 3^8 * 3^8 * 3^8 * 3^8 = 2 * 2 * 2 * 2 = 2 ( mod 7 ) 3^64 = 3^32 * 3^32 = 2 * 2 = 4 ( mod 7 )
3^105 = 3 * 2 * 2 * 4 = 6 ( mod 7 )
2. 4^8 = 4² * 4² * 4² * 4² = 2 * 2 * 2 * 2 = 2 ( mod 7 ) 4^32 = 4^8 * 4^8 * 4^5 * 4^8 = 2 * 2 * 2 * 2 = 2 ( mod 7 ) 4^64 = 4^32 * 4^32 = 2 * 2 = 4 ( mod 7 )
4^105 = 4 * 2 * 2 * 4 = 1 ( mod 7 )
3. 3^105 + 4^105 = 6 + 1 = 0 ( mod 7 )
b. divisible par 11
1. 3^8 = 3^4 * 3^4 = 4 * 4 = 5 ( mod 11 )
3^32 = 3^8 * 3^8 * 3^8 * 3^8 = 5 * 5 * 5 * 5 = 9 ( mod 11 ) 3^64 = 3^32 * 3^32 = 9 * 9 = 4 ( mod 11 )
3^105 = 3 * 5 * 9 * 4 = 1 ( mod 11 )
2. 4^8 = 4^4 * 4^4 = 9 ( mod 11 )
4^32 = 4^8 * 4^8 * 4^8 * 4^8 = 9 * 9 * 9 * 9 = 5 ( mod 11 ) 4^64 = 4^32 * 4^32 = 5 * 5 = 3 ( mod 11 )
4^105 = 4 * 9 * 5 * 3 = 1 ( mod 11 )
3. 3^105 + 4^105 = 1 + 1 = 2 ( mod 11 )
Q2.
N1. 2^125 + 1048576 = 2^128 + 2^20
factorisation : 2^20 * 3² * 11 * 43 * 211 * 281 * 331 * 86171 * 664441 * 5419 * 156492 nbre diviseurs : 21 * 3 * 2^9 = 32256 = 2016 * 16
N2. 3^125 + 3486784401 = 3^125 + 3^20 factorisation :
2²*30^20*7^2*31*43*61*211*271*547*374857981681*3454081*3369031*24151*2269 nbre diviseurs : 3*21*3*2^12 = 774144 = 2016*384
N3. 5^125 + 5^20 factorisation :
2*3²*5^20*7^2*29*43*61*127*421*449*521*4698932281*1354224218968567573270561
*15216601*7603*5296141*7621
nbre diviseurs : 3*21*3*2^14 = 3096576 = 2016*1536
