Question

If the sum $1+2+3+....+K$ is a perfect square N${}^2$ and if N is less than 100, then the possible values for K are:

• A. only 1
• B. 1 and 8
• C. only 8
• D. 8 and 49
• E. 1,8, and 49

Answer 1

Answer: (E)

1,8, and 49

S= K(K + l)/2 = $N^2$. The possible values for N${}^2$ are $1^2,2^2,3^2....{99}^2$;

For K to be integral, the discriminant 1 + 8N${}^2$ of the equation $K^2+K-2N$= 0 must be a perfect square. This fact reduces the possible values for $N^2$ to $1^2,6^2,and{35}^2$. Hence the values of K are 1, 8, and 49.

Note: There are ways of shortening the number of trials for N2 still further.but these involve a knowledge of number-theoretic theorems. The shortest way to do this problem is by testing the choices given.