SOLUTION: 1. what is the smallest integers that can be express as the sum of the square of the two positive integers in two ways? 2. fins two different numbers A and B such that a+b = (a)(b

Algebra ->  Problems-with-consecutive-odd-even-integers -> SOLUTION: 1. what is the smallest integers that can be express as the sum of the square of the two positive integers in two ways? 2. fins two different numbers A and B such that a+b = (a)(b      Log On


   

Question 464047: 1. what is the smallest integers that can be express as the sum of the square of the two positive integers in two ways?
2. fins two different numbers A and B such that a+b = (a)(b).
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
1. Suppose a,b,c,d are positive integers satisfying



Then,





After this was some "educated trial-and-error." I simply thought of numbers that would factor in two different ways (e.g. 48 = 12*4 = 24*2). The only constraints were that the factors (e.g. 12,4 and 24,2) had to be both odd or both even, and distinct, so that a,b,c,d would be nonnegative integers. I obtained 15 = 15*1 = 5*3, which gives the equations






Solving this yields a = 8, c = 7, b = 1, d = 3, and it can be checked that



which is the smallest such number (I hope).

2. Given

we can find b in terms of a,





This essentially defines a function b in terms of a, and there are infinitely many ordered pairs (a,b) that satisfy. For example, if a = 3, b = 3/2, and 3 + (3/2) = 3(3/2) = 9/2.