Question

Determinați numerele a și B știind ca îndeplinesc simultan conditiile:a)(a,b)=15, a+b=180 b)(a,b)=8, a×b=1280

Answer 1

a)  (a,b)=15, a+b=180

(a, b)=15 => a=15m si b=15n ; (m, n)=1

a+b=15m+15n

15(m+n)=180

m+n=12

(m,n)∈{(1,11), (5,7), (7,5), (11,1)}

(a,b)∈{(15, 165), (75, 105), (105, 75), (165, 15)}

b)(a,b)=8, a×b=1280

(a,b)=8 => a=8m si b=8n; (m,n)=1

a•b=8m•8n

64m•n=1280

m•n=20

(m,n)∈{(1,20), (4,5), (5, 4), (20,1)}

(a,b)∈{(8, 160), (32, 40), (40, 32), (160, 8)}