No, 16 is not the only square number that's a multiple of 4.
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sqrt(2) / x is rational only if x is a multiple of sqrt(2). sqrt(2/x) is rational only if x is two times the square of some number (not necessarily a whole number).
No, only square numbers have an odd number of factors.
Because its square root is not an integer.
2 is the only number is that both a factor of 2 and a multiple of 2.