1961 AMC 12 Problem 4

Question

Let the set consisting of the squares of the positive integers be called

u;

thus

u

is the set

1,

4,

9,

\ldots.

If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then

u

is closed under:

addition

multiplication

division

extraction of a positive integral root

none of these

Test each operation on arbitrary squares

m^2

and

n^2

The product of two squares has an immediate square form

For positive integers

m,n,

m^2n^2=(mn)^2,

which is again the square of a positive integer. The other operations fail, for example

1+4=5,\frac{1}{4},

and

\sqrt4=2.

Thus, the correct answer is B.

Problem 4 in other years