Question

Arătat ca 10 divide a , unde a=7^53+7^54+7^55+7^56-(3^27+3^28+3^29+3^30)

Arătat ca 2 divide a , unde a= 7^180+1^199+4^200+8^201
,,^" înseamnă la puterea.... Exemplu ,,7^180"
Determinati numerele naturale x , stiind ca :
9< sau egal x < sau egal 49
Va rog cât mai repede

Answer 1

$a=(7^{53}+7^{54}+7^{55}+7^{56})-(3^{27}+3^{28}+3^{29}+3^{30})$
$a=7^{53}(1+7+7^{2}+7^{3})-3^{27}(1+3+3^{2}+3^{3})$
$a=7^{53}(1+7+49+343)-3^{27}(1+3+9+27)$
$a=7^{53}\cdot400-3^{27}\cdot40$
$a=10(7^{53}\cdot40-3^{27}\cdot4)\vdots10$
$-----------------$
$a=7^{180}+1+4^{200}+8^{201}$
$O~putere~a~unui~numar~par~este~para,~iar~o~putere~a~unui~numar~impar~este~impara$
$a=impar+impar+par+par=par\vdots2$
$-----------------$
$9\leq x\leq49$
$x\in\{9,10,11,12,\dots49\}$