Find two natural numbers, the sum of whose squares is `25` times their sum and also equal to `50` times their difference.

Answer

30 and 10

Explanation

Let the required natural numbers be a and b. Then,
`a^(2)+b^(2)=25(a+b)" and "(a^(2)+b^(2))=50(a-b).`
`:." "25(a+b)=50(a-b)impliesa+b=2(a-b)impliesa=3b.`
`:." "(3b)^(2)+b^(2)=25xx4bimplies10b^(2)-100b=0`
`implies10b(b-10)=0impliesb=10.`