1. The number 10³⁰ is divisible by all of the following EXCEPT
   A. 250
   B. 125
   C. 32
   D. 16
   E. 6
2. The rectangular garden with dimensions x feet by y feet, is surrounded by a walkway 2 feet wide. Which of the following represents the area of the wlakway, in square feet?
   A. 2x + 2y + 4
   B. 2x + 2y +16
   C. 4x - 4y + 8
   D. 4x + 4y + 16
   E. 4x + 4y + 32
3. If 3x + 1 represents an odd integer, which of the following represents the larger odd integer?
   A. 3(x + 1)
   B. 3(x + 2)
   C. 3(x + 3)
   D. 3x + 2
   E. 3(x + 2) + 1
4. x - y + z = 0
   2x + y + 3z = 0
   In the system of equations above, if Z is not zero, then the ratio of x to z is
   A. -2
   B. -4/3
   C. -1/2
   D. 3/4
   E. 4/3
5. If a and b are both positive even integers, which of the following must be even?
   I. a^b
   II. (a + 1)^b
   III. a^(b+1)
   A. I only
   B. II only
   C. I and II only
   D. I and III only
   E. I, II and III
6. In the sequence of numbers a, b, c, d, e, each number after the first is twice the preceding number. If e - a is 20, what is the value of a?
   A. 4/3
   B. 5/4
   C. 2
   D. 5/3
   E. 4
7. One hour after John started walking from A to B, a distance of 45 km, Tom started walking along the same route from B to A. If John’s walking rate was 3 km/hour and Tom’s was 4 km/hour, how many kms had Tom walked when they met?
   A. 24
   B. 23
   C. 22
   D. 21
   E. 19.5
8. The average (arithmetic mean) of 6 numbers is 8.5. When one number is discarded, the average of the remaining numbers becomes 7.2. What is the discarded number?
   A. 7.8
   B. 9.8
   C. 10.0
   D. 12.4
   E. 15.0
9. If the length and width of a rectangular floor area were each increased by 30 percent, what would be the percent increase in the area of the floor?
   A. 30%
   B. 36%
   C. 60%
   D. 69%
   E. 90%
10. If n is a positive integer and k + 2 = 3ⁿ, which of the following could NOT be a value of k?
    A. 1
    B. 4
    C. 7
    D. 25
    E. 79

Answers

1. 10³⁰ = (2 x 5)³⁰ Get the point? The one with a number other that 2 and 5 cannot satisfy this.
    250 = 5 x 5 x 5 x 2 OK
    125 = 5 x 5 x5 OK and similarly until
     6     = 2 x 3  (Think for yourself how long can you make it. 1 minute? Choice E
2. Let say the walkway area = the total area (including the walkway) - the garden area
   you get the walkway area = (x +4)(y + 4) - xy  (you wold have probably added just 2 instead. Draw a sketch yourself). The correct polynomial is 4x + 4y + 16.  Choice D.
3. The next bigger odd integer than 3x + 1 is 3x + 1 + 2 = 3x +3 = 3(x + 1). Choice A. (Less than 30 seconds)
4. Trick. At the very first glance, try what exactly the question wants. Yeah, eliminating y. Then in the equation systems above, you have -y and + y. The only easiest way is to combine the two equations immediately. You’ll get; 3x + 4z = 0, so x/z = -4/3. Choice B.
5. I assume you all know such basic rules as; even x even = even blah blah blah
   I. = even x even x even............(even times) = even
   II = odd x odd x odd...............(even times) = odd
   III = even x even x even...........(odd times) = even. Choice D.
6. Your set of numbers is like this a, 2a, 4a, 8a, 16a. That is, e - a = 16a - a = 15a = 20, then a = 4/3. Choice A.
7. When they met, the distances both had made were 45 km and the time John had spent was t + 1 hr and that of Tom was t hr.
   The relationship is 3(t + 1) + 4t = 45, t = 42/7 = 6. So Tom had walked by 4 x 6 = 24 miles. Choice A.
8. The sum of the 6 numbers is 6 x 8.5 = 51 and the that of the 5 remaining numbers is 5 x 7.2 = 36. The discarded number is the difference between both sums = 15. Choice E.
9. The original area was wl  when w = width and l = length. The area after increasing is 1.3l x 1.3 w = 1.69wl, that is the area was increased by 69%
10. There is no better way to face such a problem that a substitution method.
    A. k = 1 then 1 + 2 = 3¹, n = 1
    B. k = 4 then 4 + 2 = 3^?, n is not an integer. Choice B.
    Note: It’s quite rare that the right answer for the question that ETS intends to solve by the method of substitution is Choice A or B. Usually, I recommend try reversely from E to A instead.