Studied by 11 peopleIf P is the set of multiples of 2, Q is the set of multiples of 3, and R is the set of multiples of 7, which of the following integers will be in P and Q but not in R?
A) −54 B) −50 C) 42 D) 100 E) 252
Solution:
The correct answer is A.
The correct answer will not be divisible by 7 but will be divisible by 2 and 3. This is the same as saying it is a multiple of 6 but not of 7.
Answer choice A is equivalent to −9 × 6 and is not divisible by 7, so it meets the criteria
Answer choice B is incorrect because −50 is not divisible by 6 or 7 (it is in Q).
Answer choice C is incorrect because it is divisible by both 6 and 7 (it is in P, Q, and R).
The answer is not D, 100, because 100 is not a multiple of 3 (set Q).
The answer is not E because it is divisible by both 6 and 7 (it is in P, Q, and R).
A number is a multiple of both 4 and 9. Which of the following is NOT a possible value of the number?
A) −540 B) −324 C) 126 D) 144 E) 360
Solution:
The correct answer is C.
If a number is a multiple of both 4 and 9, then it is divisible by 4 × 9 = 36. All of the answer choices except C are divisible by 36.
The remainder when an integer m is divided by 2 is 1. What is the remainder when (m + 1) is divided by 2?
A) 0 B) 1 C) 2 D) 3 E) 4
Solution:
The correct answer is A.
To review, the remainder of a division operation is the number of leftover units after the divisor is divided into the dividend to produce a quotient; the remainder is not the same as the quotient. If the remainder of m divided by 2 is 1, then m must be odd (since 2/2 = 1 (with a remainder of 0). If m is odd, then m + 1 is even and a multiple of 2. Therefore, when m + 1 is divided by 2, there will be no remainder.
What is the greatest common factor of 27, 54, and 243?
A) 3 B) 7 C) 9 D) 27 E) 243
Solution:
The correct answer is D.
Factors of 27 are 1, 3, 9, and 27. Factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. The factors of 243 are 1, 3, 9, 27, 81, and 243. The greatest factor that is common to all numbers is 27.
Which of the following lists all the positive factors of 32?
A) 1, 32 B) 2, 16 C) 2, 4, 8, 16 D) 2, 4, 8, 16, 32 E) 1, 2, 4, 8, 16, 32
Solution:
The correct answer is E.
The positive factors of 32 include all of the positive integers that divide evenly into 32. Don’t get confused between positive integers (which include 1) and even integers.
What is the least common multiple of 40, 70, and 60?
A) 240 B) 420 C) 840 d)1,680 E) 168,000
Solution:
The correct answer is C.
To solve this problem, start by finding the least common multiple of 60 and 70, which is 420. However, 420 is not a multiple of 40. Next, try 2 × 420, which is 840. The number 840 is still a multiple of 60 and 70 and is also a multiple of 40.
An approach that is very helpful (and perhaps even faster!) for this problem would be to start with the smallest answer choice (you are asked for the least common multiple) and stop when you find an answer choice that is evenly divisible by 40, 60, and 70.
If 3(m + n) is even, then which of the following must also be even?
A) m + n B) m C) n D) 3m E) 3n + 1
Solution:
The correct answer is A.
If 3(m + n) is even, then m + n must be even, since the product of two odd numbers (that is, 3 and the sum of m + n) would be odd. It is possible that both m and n are even, but it is also possible that they are both odd. This eliminates the remaining answer choices.
What is the lowest possible value for the product of 2 integers that differ by 7?
A) 8 B) 0 C) −6 D) −10 E) −12
Solution:
The correct answer is E.
The lowest possible value will occur when it is negative. A negative product will result only when one of the numbers is positive and one is negative. The possible pairs are then −1 and 6, −2 and 5, −3 and 4, −4 and 3, −5 and 2, and −6 and 1. Of these pairs, the smallest product is (−3)(4) = (−4)(3), or −12.
If m, n, and p are positive integers such that m + n is even and the value of (m + n)2 + n + p is odd, which of the following must be true?
A) m is odd B) n is even C) p is odd D()If n is even, p is odd E) If p is odd, n is odd
Solution:
The correct answer is D.
You are given that m + n is even and the value of (m + n)2 is also even. However, because (m + n)2 + n + p is odd, the sum n + p must be odd. A sum of two positive integers is odd only when one is even and one is odd. Therefore, it must be true that if n is even, p is odd.
What is the smallest possible value for the product of 2 real numbers that differ by 6? (Assume neither number is 0.)
A) −9 B) −8 C) −5 D) 0 E) 7
Solution:
The correct answer is A.
If 2 numbers, x and y, differ by 6, that means that x − y = 6. Multiplying the two numbers, (x)(y), will yield the product. Solve the first equation for x.
x−y=6x=y+6 Substitute the result for x in the second equation.
(y+6)y Since one of the answer choices must be the solution to that equation, plug in the answer choices, starting with the smallest value (note that the answer choices are in ascending order):
(y+6)y=−9y2+6y+9=0(y+3)2=0y=−3 Now, substitute −3 for y in the first equation and solve for x:
x−(−3)=6x=3 Since (x)(y) = (3)(−3) = −9 is the smallest value given as an answer, answer choice A must be correct.
Note
Flashcard