Răspuns:
Explicație pas cu pas:
2^2n-2 = (2^2)^n-1 = 4^n-1
4^n-1*7^n-1 = (4*7)^n-1 = 28^n-1
_________
4^n*7^n = (4*7)^n = 28^n
2^2n+3 = 2*2^2n+2 = 2*(2^2)^n+1 = 2*4^n+1
2*4^n+1*7^n+1 = 2*28^n+1
__________
A = 35*224*28^n-1 - 81*28^n - 3*2*28^n+1
= 7840*28^n-1 - 81*28^n - 6*28^n+1
7840 = 28*28*10 = 10*28^2
A = 10*28^2*28^n-1 - 81*28^n - 6*28^n+1
= 10*28^n+1 - 81*28^n - 6*28^n+1
= 4*28^n+1 - 81*28^n
= 4*28*28^n - 81*28^n
= 112*28^n - 81*28^n
= 31*28^n este divizibil cu 31
____________________
31*28^n = 14^3*248
14^3 = 2^3*7^3
248 = 31*8 = 31*2^3
14^3*248 = 2^3*7^3*31*2^3 = 31*2^6*7^3 = 31*4^3*7^3 = 31*28^3
31*28^n = 31*28^3
n = 3
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Răspuns:
Explicație pas cu pas:
2^2n-2 = (2^2)^n-1 = 4^n-1
4^n-1*7^n-1 = (4*7)^n-1 = 28^n-1
_________
4^n*7^n = (4*7)^n = 28^n
_________
2^2n+3 = 2*2^2n+2 = 2*(2^2)^n+1 = 2*4^n+1
2*4^n+1*7^n+1 = 2*28^n+1
__________
A = 35*224*28^n-1 - 81*28^n - 3*2*28^n+1
= 7840*28^n-1 - 81*28^n - 6*28^n+1
7840 = 28*28*10 = 10*28^2
A = 10*28^2*28^n-1 - 81*28^n - 6*28^n+1
= 10*28^n+1 - 81*28^n - 6*28^n+1
= 4*28^n+1 - 81*28^n
= 4*28*28^n - 81*28^n
= 112*28^n - 81*28^n
= 31*28^n este divizibil cu 31
____________________
31*28^n = 14^3*248
14^3 = 2^3*7^3
248 = 31*8 = 31*2^3
14^3*248 = 2^3*7^3*31*2^3 = 31*2^6*7^3 = 31*4^3*7^3 = 31*28^3
31*28^n = 31*28^3
n = 3