Find the number of ordered triples (a, b, c) of positive integers such that 30 a+50 b+70 c 343 .

Step-by-step solution

aligned & 30 a+50 b+70 c 343 & 3 a+5 b+7 c 34.3 & 3 a+5 b+7 c 34 & a, b, c N, a=1+p, b=1+q, c=1+r & 3 p+5 q+7 r 19 & If r=0, & q=0 p can take 7 values & q=1 p can take 5 values & q=2 p can take 4 values & q=3 p can take 2 values & 18 values & If r=1 then 3 p+5 q 12 & q=0 p can take 5 values & q=1 p can take 3 values & q=2 p can take 1 values & 9 values & If r=2 then 3 p+5 q 5 & q=0 p can take 2 values & q=1 p can take 1 value & 3 values & Total 18+9+3=30 values & 30 a+50 b+70 c 343 & 3 a+5 b+7 c 34.3 & 3 a+5 b+7 c