X = a.bc
If a, b and c denote the units, tenths and hundredths digits in the decimal representation of X above, what is the value of the product abc?
(1) 100X divided by 50 leaves a remainder 24
(2) 1000X divided by 8 leaves a remainder 0
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
Statement 1 is not sufficient to answer the question.
Step 4: Analyze Statement 2 independently
Therefore, Statement 2 is clearly not sufficient
Step 5: Analyze Both Statements Together (if needed)
So even after combining both statements, we still don’t know the definite values of a and b, and therefore cannot find a unique value of the product a×b×c
The two statements together are not sufficient to answer the question.
Answer: Option E
If z is a positive integer and r is the remainder when z^{2} + 2z + 4 is divided by 8, what is the value of r?
(1) When (z-3)^{2} is divided by 8, the remainder is 4
(2) When 2z is divided by 8, the remainder is 2
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: The value of r
Step 3: Analyze Statement 1 independently
Thus, Statement 1 alone is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
Thus, Statement 2 alone is sufficient to find a unique value of the remainder
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D
If x is an integer that lies between 390 and 400, exclusive, what is the value of x?
Steps 1 & 2: Understand Question and Draw Inferences
Given: x = {391, 392, 393 . . . 398, 399}
To find: x = ?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘x^{3} has the same units digit as x’
Since Statement 1 leads us to 5 possible values of x, it is clearly not sufficient to get us a unique answer
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘x + 2 is divisible by 9’
Statement 2 is sufficient to get us a unique answer
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already got a unique answer in Step 4, this step is not required
Answer: Option B
For a non-negative integer n, the function mod(n, d) denotes the remainder obtained when n is divided by positive integer d. Which of the following statements must be true?
I. mod(n, d) = mod(-2n, d)
II. If mod(n, d) = 1, then mod(5n, d) = 5
III. [mod(n, d)]^{2 } = mod (n^{2}, d)
Given:
To find: Which of the 3 statements must be true?
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option E
When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?
Given:
To find: The smallest number that can be added to or subtracted from n to make the resulting number divisible by 12
Approach:
1. Let the number to be added or subtracted from n be x. This means,
2. The expressions n = 12j – x and n = 12m + x make us realize that in order to answer this question, we need to know the remainder when n is divided by 12
3. Let the remainder that n leaves when divided by 12 be r. So, our Goal expression is: n = 12q + r, where quotient q is an integer and 0 ≤ r < 12
4. We’ll use the given information about n to drive towards our Goal expression
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
If x is a positive integer and f(x) = x – x^{2} – x^{3} + x^{4} + x^{5} – x^{6} – x^{7} + x^{8 }, then is f(x) divisible by 96?
(1) x – x^{2} – x^{3} is divisible by 32
(2) x has no prime factors other than 2
Steps 1 & 2: Understand Question and Draw Inferences
Given:
f(x) = x – x^{2} – x^{3} + x^{4} + x^{5} – x^{6} – x^{7} + x^{8}
= x(1-x) – x^{3}(1-x) +x^{5}(1-x) – x^{7}(1-x)
=(x – x^{3} + x^{5} – x^{7})(1-x)
=[x(1-x^{2}) +x^{5}(1-x^{2})](1-x)
=x(1+x^{4})(1-x)(1+x)(1-x)
To find:
The expression for f(x), that is (x-1)^{2}(x)(x+1)(x^{4}+1), contains the product of 3 consecutive integers ( x-1, x and x+1 ).Therefore, f(x) is definitely divisible by 3 (because in any set of 3 consecutive integers, one will definitely be a multiple of 3)
Let’s evaluate the conditions in which f(x) will be divisible by 2^{5}
The divisibility of f(x) by powers of 2 will be impacted by the even-odd nature of x
Case 1: x is even
Case 2: x is odd
Step 3: Analyze Statement 1 independently
Case 2: x is odd
Therefore, Statement 1 is sufficient.
Step 4: Analyze Statement 2 independently
Thus, Statement 2 alone is not sufficient to arrive at a definite answer to the question.
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
Set A = {20, P, 80}
Set B = {1, 3, 5, 23}
When the mean of the Set A is divided by the median of the Set B, the result is 15.25. What is the remainder when P^{4} is divided by the mean of the Set B?
Given:
To find: The remainder when P^{4} is divided by the mean of Set B
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option A
(Note: To get more practice on Binomial Theorem, you may look at the Divisibility and Remainders Diagnostic Test – 2 questions in that test make use of the Binomial Theorem and their solutions contain a detailed explanation of Binomial Theorem. If you are a Quant Live Prep student, you may also look at the Divisibility questions in Number Properties 2 Live Session)
The arithmetic mean (average) M of 4 terms is an integer. When M is divided by 16, the remainder is 10. If each of the terms is increased by 100%, what is the remainder when the new mean is divided by 16?
Given:
To find: The remainder when the new mean is divided by 16
Approach:
2. Since we’re given the relation M = 16q + 10, we’ll express M in terms of M’ and thus get an expression for M’ in the form 16k + r
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
The students of a class are to be arranged in rows, starting from number 1, such that a row can have a maximum of x students, where x is a number to be determined that is greater than 1. If the minimum number of rows required to accommodate all the students is 10, what is the number of students in the class?
(1) If 3 students are shifted from row number 9 to row number 10, both the rows would have an equal number of students.
(2) Had there been 1 student less in the class, the minimum number of rows required would have been 9.
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Since we are talking about minimum number of rows
Two cases are possible:
We need to find the values of x and z.
Step 3: Analyze Statement 1 independently
Statement I conveys that a shifting of students is required to achieve an equal number of students in Rows 9 and 10. This means that the number of students in these two rows initially is not equally. So, what we know for sure now is that Case a is rejected. It is Case-b that is applicable.
As per Case b, Number of students in 9^{th} row = x
Number of students in 10^{th} row = z
As per Statement 1, x -3 = z + 3
So, x = z +6.
2 unknowns and 1 equation. We cannot find unique values of x and z. Insufficient to answer the question
Step 4: Analyze Statement 2 independently
(2) Had there been 1 student less in the class, the minimum number of rows required would have been 9.
This means, there is only 1 student in the last row. Case-b is applicable.
z = 1. However, it does not tell us about the value of x. Insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
St- 1 gives us relation between x and z and St-2 gives us the value of z. Thus, we can find a unique value of x.
Sufficient to answer the question
Answer: C
A function D(a, 10b + c) is defined as the remainder when the sum a^{0 }+ a^{1}….a^{10b + c} is divided by c, where a, b and c are single-digit positive integers.
What is the value of D(y, 10x + z) where x, y and z are single-digit positive integers such that x < y < z , x and z are perfect squares and the difference between the sum and the product of the prime factors of y is 1?
Given
To Find: Value of D(y, 10x + z)
Approach
Working Out
3. Let’s find the value of D(y, 10x + z) for both the cases of values of (x, z)
a. Case-I: y = 6 and (x, z) = (1, 9)
b. Case-II: y = 6 and (x, z) = (4, 9)
4. We see that in both the cases, the value of D(x, 10y+z) = 7.
Answer: D
The arithmetic mean (average) of a set of 5 numbers is an integer A. Upon being divided by 180, A leaves a remainder of 30. If two of the numbers in the set are decreased by 120 and 180 units respectively, which of the following statements must be true about the new arithmetic mean B of the set?
I. The remainder that B leaves upon division by 180 is less than the remainder that A leaves upon division by 180
II. When B is divided by 60, the remainder is 30
III. The value of the decimal digits is greater in the number than in the number
To find: Which of the 3 statements is/are true?
Approach:
Working Out:
Thus, B = A – 60
Getting to the answer
Looking at the answer choices, we see that the correct answer is Option B
If x is a positive integer, what is the remainder when 15^{x} + x^{15} is divided by 4?
(1) The sum of the product of x and z and the sum of x and z is even, where z is a positive integer.
(2) y when divided by x is equal to 13.45, where y is a positive integer
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To Find: The remainder when 15^{x} + x^{15} is divided by 4
Combining the analysis of both the terms:
Statement-1 is sufficient to answer the question
Step 4: Analyze Statement 2 independently
(2) y when divided by x is equal to 13.45, where y is a positive integer
Statement-2 is sufficient to answer the question
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps-3 and 4, we do not need this step
Answer: D
If x is a prime number, which of the following statements must be true?
Given: Prime number x
To find: Which of the statements must be true?
Approach:
Working Out:
So, the correct answer is Option A
The expression 2a^{2} +2b^{2} can be written in the form of 18x + y, where a, b, x and y are non-negative integers and y < 18. Is |y+3| = 3?
(1) The difference between a and b can be expressed as an even multiple of 3.
(2) b when divided by 3 is an integer.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: If |y+3| = 3?
Step 3: Analyze Statement 1 independently
Insufficient to answer.
Step 4: Analyze Statement 2 independently
2. b when divided by 3 is an integer.
Hence, insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Suffecient to answer.
Answer: C
If k and m are positive integers, what is the remainder when k is divided by m +1?
(1) k-m when divided by m+1 leaves a remainder 1
(2) k/3 and k/2 when divided by m+1 leave a remainder equal to 4 and 3 respectively.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: remainder when k is divided by m +1?
Step 3: Analyze Statement 1 independently
Sufficient to answer
Step 4: Analyze Statement 2 independently
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
If n is a perfect square of a positive integer and is an integer, then
Given
To Find: To check the divisibility of in the given options
Approach
Working Out
2. Evaluating the given options:
Answer: C
If x and y are positive integers such that x/y=5.4, which of the following could be the value of y?
Given: Integers x, y > 0
x/y=5.4,
To find: The possible value of y
Approach:
Working Out:
We are given that x is an integer.
Correct Answer – Option D
If x is a positive integer such that x - 1 and x + 6 are not divisible by 3, which of the following must be divisible by 3?
Given: Integer x > 0
To find: Which of the given 5 terms is divisible by 3?
Approach:
Working Out:
Looking at the answer choices, we see that (x + 7) is one of the answer choices. So that is the correct answer – Option D.
If T is a prime number, what is the remainder when T is divided by 3?
(1) (T – 13)^{3} is divisible by 48
(2) 4T – 3 leaves a remainder 1 when divided by 18
Steps 1 & 2: Understand Question and Draw Inferences
Given: Prime Number T
To find: Remainder when T is divided by 3
Step 3: Analyze Statement 1 independently
Statement 1 alone is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
Thus, Statement 2 alone is also sufficient to obtain a unique answer
Step 5: Analyze Both Statements Together (if needed)
Since we’ve arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D
Is the square of positive integer Z divisible by 9?
(1) The sum of the digits of Z^{3} is divisible by 9
(2) 3Z^{4} + 16 leaves a remainder of 7 when divided by 9
Steps 1 & 2: Understand Question and Draw Inferences
Given: Positive integer Z
To find:
Step 3: Analyze Statement 1 independently
Hence, Z is indeed divisible by 3
Statement 1 is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
So, Z is indeed divisible by 3
Statement 2 alone is sufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D
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