n is a positive integer.
The remainder when 5n is divided by 4 is 3
| Quantity A | Quantity B |
| The remainder when 10n is divided by 4 | 2 |
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Solution
If the remainder is 3, then 5n must be 3 more than a multiple of 4, such as 4, 8, 12, or 16.Try adding 3 to these multiples to find a possible value for 5n. 12 + 3 yields 15 as a value for 5n; n = 3. Quantity A is the remainder when 30 is divided by 4, or 2.Eliminate choices (A) and (B).Try a different number. If n is 7, then 5n is 35, which also has a remainder of 3 when divided by 4. In Quantity A, 70 divided by 4 has a remainder of 2. For any other numbers you try, choice (C) will be the answer.
If x and y are letters which correspond to points on the number line shown above, which of the following statements must be true?
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Solution
Plug in values for x and y that fit the figure:Try x = 1⁄6 and 2⁄3. Now, plug these numbers into each of the choices and use POE. Only choice (D) is correct:1⁄6×2⁄3=2⁄18=1⁄9, which is less than 1⁄3.
y = |y|
y = −|y|
| Quantity A | Quantity B |
| y | 0 |
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Solution
The first equation tells you that y cannot be a negative number.The second equation tells you that y cannot be a positive number. Therefore, y must be 0.
If a and b are integers, ab = −5, and a − b > 0, which of the following must be true?
I. a > -1
II. b is odd
III. |a| = 5
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Solution
If a and b are integers with a product of −5, then there are only 4 options: a = 5 and b = −1; a = −5 and b = 1; a = 1 and b = −5; and a = −1 and b = 5.The requirement that a − b > 0 eliminates the second and fourth options, leaving only a = 5 and b = −1 and a = 1 and b = −5. (I) and (II) are both true for these two cases and (III) is not true if a = 1, making choice (C) the answer.
If p and q are both positive odd integers, which of the following must be odd?
Indicate all possible values.
| A.pq |
| B.2pq |
| C.3pq |
| D.pq + pq |
| E.pq + qp |
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Solution
A and C
As soon as you see variables in the answer choices, set up your scratch paper to Plug In. Start with easy numbers like p = 3 and q = 5, and eliminate any answer choice that doesn’t yield an odd result.Choice (A) is 15, so keep it.Choice (B) is 30, so eliminate choice (B).Choice (C) is 45, so keep it.Choice (D) is 258, and Choice (E) is 368, so you can eliminate both; if you recognize them as the sum of two odd numbers, you don’t have to calculate either of them. It’s a must be problem, so try another set of numbers in choices (A) and (C) to be sure; as long as p and q are both positive odd integers, choices (A) and (C) will always work.
How many positive integers less than 20 are factors of 96?
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Solution
The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.Eight of these numbers are less than 20.
f, g, and h are consecutive prime numbers such that f < g < h.
| Quantity A | Quantity B |
| f + g + h | 3g |
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Solution
The intervals between consecutive prime numbers does not follow a consistent, predictable pattern. Prove it by Plugging In:Try f = 2, g = 3, and h = 5. Now f + g + h = 10 and 3g = 9. Quantity A is greater; eliminate choices (B) and (C). Now try f = 7, g = 11, and h = 13.This time, f + g + h = 31, and 3g = 33. Quantity B is now greater.Eliminate choice (A), and you’re left with choice (D).
which of the following integers has both 12 and 17 as factors?
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Solution
Eliminate choices (A) and (E) because they are not divisible by 12.Eliminate choices (B) and (C) because they are not divisible by 17.
If x, y, and z are consecutive even integers, such that x < y < z and xyz = 960, what is the value of z?
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Solution
12
Ballpark that 960 is about 1,000, which is 10 × 10 × 10.Then test a set of consecutive even integers near 10, such as 10 × 12 × 14 = 1,680.This product is too large.Try 8 × 10 × 12 = 960, giving you z = 12.
If r is an integer multiple of 8, then which of the following could NOT be divisible by r?
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Solution
Plug in a value for r:The first integer multiple of 8 is 8 itself. Only choice (E) fails to yield an integer: \(\frac{420}{8}=52.5\)