GRE math questions
Which of the following expressions are *factors* of 12^17? Indicate all possible correct answers. a. (-2)^2(-3)^4 b. 12^3 + 12^3 c.4^18 d. 2^16× 3^15 e. 3^17× 4^17 f. 12^17 + 12^17
Answer: The correct answers are answer choices *(A), (B), (D), and (E).* The easiest way to find expressions that are factors of 1217 is through prime factorization. The prime factorization of 12 is 2 × 2 × 3. 1217 simply means there are thirty-four 2s and seventeen 3s, so any choice that does not need something other than that is a factor. (A) (-2)2(-3)4 = 22× 34. That is two 2s and four 3s, so this is a factor. (B) 123 + 123 = 2 × 123 = 2(2 × 2 × 3)3 = 27× 33. Seven 2s and three 3s, so this is a factor. (C) 418 = 236. Thirty-six 2s is too many because 1217 only has thirty-four 2s. Eliminate. (D) 216× 315. Sixteen 2s and fifteen 3s, so this is a factor. (E) 317× 417 = 317× 234. Exactly seventeen 3s and thirty-four 2s. This is a factor. (F) 1217 + 1217 = 2 × 1217. This is larger than 1217, so it cannot be a factor. Eliminate.
If the integer m is a multiple of both 10 and 12, then all of the following must be factors of m EXCEPT 15 12 10 8 6
Checking the choices, we see that only 8 is not a factor of 60. So (D) must be correct. We can pick numbers. Because we are told that all of the answer choices except for one "must be factors of m," we know that an answer choice that isn't a factor of m will be the correct answer. Let's choose a value or m that is permissible. We need a multiple of 10 and 12. Let's choose 60. Let's check each of the answer choices to determine whether any of the choices aren't factors of 60.
p2 + 3p + 2 = 0 Quantity A: p Quantitiy B: 1 Compare the two quantities and select the appropriate answer. Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.
The correct answer is B. This is a great illustration of the payoff on Test Day for spending time now to learn how quadratics work. No calculation is required by this question if we can look at the given quadratic and realize that both its roots will be negative, so Quantity B must be greater. Look at the "+2" in the quadratic expression. What does it signify? It implies that the signs on the second terms of the binomials into which the quadratic expression can be factored must match. Now look at the "+3 p " in the quadratic. What does it mean? It means that the signs will both be positive. So if the second terms of the binomials are positive, the roots themselves — the values of p that the quadratic ultimately solves for — will be negative:
g and h are consecutive multiples of 4. Quantity A: 5(g - h)2 Quantity B: 80 Compare the two quantities and select the appropriate answer. Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.
The correct answer is C. If g and h are consecutive multiples of 4, the absolute difference between g and h will always be 4, therefore . The quantity in Quantity A is 80.
AB = 17, AD = 8, and BC = 9. Quantity A: The volume of the rectangular solid Quantity B: 864 Compare the two quantities and select the appropriate answer. Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.
The correct answer is C. Let's connect points B and D with a line segment. Now triangle BCD is a right triangle. We have (BD)2 = (BC)2 + (CD)2. Triangle ABD is a right triangle. We have (AB) 2 = (AD)2 + (BD)2. Using the equation (BD)2 = (BC)2 + (CD)2 that we obtained from right triangle BCD, let's substitute (BC)2 + (CD)2 for (BD)2 in the equation (AB) 2 = (AD) 2 + (BD)2, which is the equation that we obtained from right triangle ABD. Then we have (AB)2 = (AD)2 + (BC)2 + (CD)2. We know that AB = 17, AD = 8, and BC = 9. The only quantity that we do not know in the equation (AB)2 = (AD)2 + (BC)2 + (CD)2 is CD. We can find the length of CD by substituting AB = 17, AD = 8, and BC = 9 in the equation (AB)2 = (AD)2 + (BC)2 + (CD)2. We have 172 = 82 + 92 + (CD)2. Then 289 = 64 + 81 + (CD)2, 289 = 145 + (CD)2, and 144 = (CD)2. Now lengths cannot be negative, so . The volume of any rectangular solid is length times width times height. The volume of this rectangular solid is 12 × 9 × 8 = 108 × 8 = 864. The quantities in both quantities are equal to 864 and choice (C) is correct.
The average (arithmetic mean) of the numbers v, w, x, y, and z is j, and the average of the numbers x, y, and z is k. What is the average of v and w in terms of j and k ? (j - k) / 2 (j + k) / 2 (5j - 3k) / 3 (5j - 3k) / 2 (5j - k) / 2
The correct answer is D. The average formula is . We can use this formula in the rearranged form Sum of the terms = (Average) × (Number of terms). Since the average of v, w, x, y, and z is j, the sum of the 5 terms v, w, x, y, and z is 5j. Since the average of x, y, and z is k, the sum of the 3 terms x, y, and z is 3k. If we subtract from the sum of v, w, x, y, and z the sum of x, y, and z, we will be left with the sum of v and w. The sum of v and w is 5j − 3k, that is, v + w = 5j − 3k. The average of the 2 terms v and w is .