Question

aflati nr a si b stiind ca (a,b)×[a,b]=2^p×3^q×5^r unde p,q,r apartne lui N fara 0 ..p<q<r si a×b are 60 divizori intregi

Answer 1

(a,b)×[a,b]=2^p×3^q×5^r; p,q,r∈N*; p<q<r; a•b are 60 divizori intregi

=> a•b are 30 divizori naturali

a•b= (a,b)×[a,b]=2^p×3^q×5^r

nr divizorilor naturali=(p+1)(q+1)(r+1)=30

(p+1)(q+1)(r+1)=2•3•5 si p<q<r =>

p=1; q=2; r=4;

a•b=2•3^2•5^4=11250

(a,b)=d si [a,b]=m ; m=d•t, t∈N

d•m=11250; d•d•t=2•3^2•5^4=2•(3•5^2)^2 => d=3•5^2=75

(a,b)=75 ; => a=75•x si b=75•y;   (x,y)=1

=> a•b=75•x•75•y=11250

=> x•y=2

(x,y)={(1,2), (2,1)}

(a,b)={75, 150), (150,75)}