Question

determinati m si n stiind ca [m,n]-(m,n)=47

Answer 1

[m,n]-(m,n)=47

notam (m,n)=d

[m,n]=x•d, x∈N

=> x•d-d=47

d(x-1)=47 =>

1. d=47 si x-1=1; x=2 sau

2. d=1 si x-1=47; x=48

1.   d=47 si x=2

(m,n)=47 si [m,n]=2•47=94

=> m•n=47•94

m=47•y si n=47•z; (y,z)=1

m•n=47•y•47•z=47•94

=> y•z=2 => (y,z)∈({(1,2), (2,1)}

(m,n)∈{(47,94), (94,47)}

2. d=1 si x=48

(m,n)=1 si  [m,n]=1•48=48

48=1•48=3•16; m si n sunt prime intre ele

=> (m,n)∈{(1,48), (3,16), (16,3), (48,1)}

(m,n)∈{(1,48), (3,16), (16,3), (47, 94),  (48,1), (94,47)}