Set P consists of the first n positive multiples of 3 and set Q consists of the first m positive multiples of 5. The sum of all the numbers in set P is equal to R and the sum of all the numbers in set Q is equal to S. If m and n are positive integers, is the difference between R and S odd?
(1) m is odd and n is even
(2) m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x, where x is a positive integer
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
(1) m is odd and n is even
Step 4: Analyze Statement 2 independently
(2) m can be expressed in the form of 4x +3 and n can be expressed in the form of 2x, where x is a positive integer
Step 5: Analyze Both Statements Together (if needed)
(1) From statement1, we know that is odd and n is even
(2) From statement2, we know that R may be even or odd and S is even
Answer: E
If a,b and c are three positive integers such that at least one of them is odd, which of the following statements must be true?
Given
To Find: the option that must be true (for all possible values of a,b and c)
Approach
Working Out
Hence b/2+c/2 will be odd.
So, optionIV is not always true.
As none of the options are true, the correct answer will be option E
Answer: E
If n is an integer, the number of integers between, but not including, n and 3n cannot be
I. 13
II. 14
III. 15
Given:
To find: Which of the following CANNOT be the number of integers between, but not including, n and 3n: {13, 14, 15}
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
If z is an integer greater than 1, is z even?
(1) 2z is not a factor of 8
(2) 3z/4 is a factor of 6
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integer z > 1
To find: Is z even?
Step 3: Analyze Statement 1 independently
Thus, Statement 1 is not sufficient to answer the question
Step 4: Analyze Statement 2 independently
Statement 2 is sufficient to answer the question
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
Set A consists of a set of n consecutive integers. Is the sum of all the integers in set A even?
(1) If 5 is added to set A, the set would become symmetric about 0
(2) If the largest integer of set A is removed, the sum of the remaining integers is even
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
(1) If 5 is added to set A, the set would become symmetric about 0
Step 4: Analyze Statement 2 independently
(2) If the largest integer of set A is removed, the sum of the remaining integers is even.
Step 5: Analyze Both Statements Together (if needed)
Answer: A
If A and B are integers, is the product AB even?
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integers A and B
To find: Is the product AB even?
Step 3: Analyze Statement 1 independently
Case 1: B is even
Case 2: B is odd
This means, the product AB is definitely even.
Statement 1 is sufficient to answer the question
Step 4: Analyze Statement 2 independently
13A^{2 } AB =Even
Step 5: Analyze Both Statements Together (if needed)
If A and B are positive integers, is A – B even?
1. The product of A and B is even
2. A + B is odd
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integers A, B > 0
To find: Is A – B even?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The product of A and B is even’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘A + B is odd’
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
If P and Q are positive integers, then is (P+2)(Q1) an even number?
(1) p/3Q is an even integer
(2) is a positive odd integer
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integers P, Q > 0
To find: Is (P+2)(Q1) even?
Step 3: Analyze Statement 1 independently
Thus, Statement 1 is sufficient to answer the question
Step 4: Analyze Statement 2 independently
Therefore, Statement 2 is not sufficient.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 3, this step is not required
Answer: Option A
There are N students in a class. When the students are distributed into groups that contain 4A number of students each, 3 students are left without a group. When the students are distributed into groups that contain A/3 number of students each, no students are left without a group. Which of the following statements is correct?
I. If the students are distributed into groups that contain A+ 1 students each, the number of students that are left without a group can be 2
II. If the students are distributed into groups that contain 3 students each, no students are left without a group
III. If the students are distributed into groups that contain 12 students each, 9 students are left without a group
Given:
To find: Which of the 3 statements is/are correct?
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
Which one of the following is even?
Step 1:
5 × 7 = 35
1 × 3 = 3
9 × 5 = 45
Step 2:
Only 3 × 4 = 12 is even. Rest are odd.
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