GRE Exam Numerical Ability Number System

The Euclidean algorithm is used to find the greatest common divisor (gcd) of two positive integers a and b .
input(a)
input(b)
while b>0
begin
r := a mod b
a := b
b := r
end
gcd := a
output(gcd)
When the algorithm is used to find the greatest common divisor of a = 273 and b = 110, which of the
following is the sequence of computed values for r ?
(A) 2, 26, 1, 0
(B) 2, 53, 1, 0
(C) 53, 2, 1, 0
(D) 53, 4, 1, 0
(E) 53, 5, 1, 0

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GRE Other Question

Suppose that two binary operations, denoted by + and * are defined on a nonempty set S, and that the
following conditions are satisfied for all x, y, and z in S:
(1) x+y and x * y are in S
(2) x+(y+z) = (x+y)+z and x*(y*z) = (x*y)*z
(3) x+y = y+x
Also, for each x in S and for each positive integer n, the elements nx and n x are defined recursively as x^n are defined recursively as follos:
1x = x^1 = x and
If kx = x^k have been defined, then (k+1)x = kx + x and x^(k+1) = x^k * x.
Which of the following must be true?
I. (x*y)^n = x^n * y^n for all x and y in S and for each positive integers n.
II. n(x+y) = nx ny for all x and y in S and for each positive integer n.
III. x^m * x^n = x^(m+n) for each x in S and for all positive integers m and n.
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III The minimal distance between any point on the sphere (x-2)^2 + (y-1)^2 + (z-3)^2 = 1 and any point on the sphere (x+3)^2 + (y-2)^2 + (z-4)^2 = 4 is
(A) 0
(B) 4
(C) Sqrt(27)
(D) 2(Sqrt2 + 1)
(E)3(Sqrt3 – 1)