If x is a positive integer, is ~$\sqrt{x}$~ an integer?

(1) ~$\sqrt{4x}$~is an integer.

(2)~$\sqrt{3x}$~ is not an integer

John took a test that had 60 questions numbered from 1 to 60. How many of the questions did he answer correctly?

(1) The number of questions he answered correctly in the first half of the test was 7 more than the number he answered correctly in the second half of the test.

(2)He answered ~$5 \over 6$~ of the odd-numbered questions correctly and ~$4 \over 5$~ of the even-numbered questions correctly.

If x and y are integers, what is the value of y ?

(1) xy = 27

(2) x = y²

Each Type A machine fills 400 cans per minute, each Type B machine fills 600 cans per minute, and each Type C machine installs 2,400 lids per minute. A lid is installed on each can that is filled and on no can that is not filled. For a particular minute, what is the total number of machines working?

(1)A total of 4,800 cans are filled that minute.

(2) For that minute, there are 2 Type B machines working for every Type C machine working.

If r, s, and t are nonzero integers, is r⁵s³t⁴ negative?

(1) rt is negative.

(2) s is negative.

In the xy-plane, point (r,s) lies on a circle with center at the origin. What is the value of r² + s² ?

(1) The circle has radius 2.

(2) The point(~$\sqrt{2}$~,-~$\sqrt{2}$~) lies on the circle.

A collection of 36 cards consists of 4 sets of 9 cards each. The 9 cards in each set are numbered 1 through 9. If one card has been removed from the collection, what is the number on that card?

(1) The units digit of the sum of the numbers on the remaining 35 cards is 6.

(2) The sum of the numbers on the remaining 35 cards is 176.

If x, y, and z are three integers, are they consecutive integers?

(1) z - x = 2

(2) x < y < z

If rs ≠0, is~$1 \over r$~ +~$1 \over s$~=4 ?

(1) r + s = 4rs

(2) r = s

If y = x² - 6x + 9, what is the value of x ?

(1) y = 0

(2) x + y = 3