GMAT Exam > GMAT Questions > If f(x) = x + x2, is f(a+1) – f(a) divi...
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If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an integer > 0
(1) f(a) is divisible by 4
(2) (-1)a < (-1)a+1
- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an...
Steps 1 & 2: Understand Question and Draw Inferences
The question wants us to know whether f(a+1) – f(a) is divisible by 4. Let’s simplify the expression given to us.
f(a+1) – f(a) = (a+1) + (a+1)2 – (a + a2)
Simplifying we get f(a+1) – f(a) = (a + 1 –a) + ((a+1)2 -a2))
- 1 + (a + 1 –a)(a+1+a) = 2 + 2a = 2(1+a) … (using a2 – b2 = (a-b)(a+b))
- From the statement above we can conclude that the given expression is always divisible by 2.
Hence, for f(a+1) – f(a) to be divisible by 4, (1+a) must be divisible by 2, which means that a must be odd.
Step 3: Analyze Statement 1
Statement 1 says that f(a) is divisible by 4.
f(a) = a(1+a)
a(1+a) is the product of two consecutive integers. Therefore, one term out of a and 1+a will be even and the other will be odd. The product of these two terms will be even and will always be divisible by 2.
But, we are given that a(1+a) is divisible by 4 also.
This can happen only if
a) a is divisible by 4 or
b) 1+a is divisible by 4 or
c) Both a and 1+a are divisible by 2
Case c) is ruled out since a and 1+a are consecutive terms. Therefore, they cannot be both even.
If a is divisible by 4, then a is even.
If 1+a is divisible by 4, then a is odd.
Thus, we cannot determine with confidence whether a is odd or not.
Since Statement 1 does not give us a unique answer, it is not sufficient.
Step 4: Analyze Statement 2
Statement 2 says that (-1)a < (-1)a+1
This is only possible if a is odd, implying that a+1 is even.
Thus, a is indeed odd.
Since statement 2 gives us a unique answer, it is sufficient to arrive at the conclusion.
Step 5: Analyze Both Statements Together (if needed)
Since statement 2 gave us a unique answer, this step is not needed.
Correct Answer: B
Most Upvoted Answer
If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an...
**Statement (1) alone**: f(a) is divisible by 4.
In order to determine if f(a+1) - f(a) is divisible by 4, we need to understand the function f(x) = x * x^2.
Let's substitute a into the function:
f(a) = a * a^2 = a^3
So, statement (1) tells us that a^3 is divisible by 4.
To determine if f(a+1) - f(a) is divisible by 4, we can simplify the expression:
f(a+1) - f(a) = (a+1) * (a+1)^2 - a * a^2
= (a+1) * (a+1) * (a+1) - a^3
= (a+1)^3 - a^3
To check if (a+1)^3 - a^3 is divisible by 4, we can factorize it using the binomial theorem:
(a+1)^3 - a^3 = a^3 + 3*a^2 + 3*a + 1 - a^3
= 3*a^2 + 3*a + 1
Now, if we substitute a = 0 into the expression, we get:
3*(0)^2 + 3*(0) + 1 = 1
Since 1 is not divisible by 4, we can conclude that statement (1) alone is not sufficient to determine if f(a+1) - f(a) is divisible by 4.
**Statement (2) alone**: (-1)^a * (-1)^(a+1)
Statement (2) can be simplified as (-1)^a * (-1)^(a+1) = (-1)^a * (-1)^a * (-1) = (-1)^(2a) * (-1)
= 1 * (-1) = -1
This means that (-1)^a * (-1)^(a+1) is always equal to -1, regardless of the value of a. Since -1 is not divisible by 4, we can conclude that statement (2) alone is not sufficient to determine if f(a+1) - f(a) is divisible by 4.
**Statements (1) and (2) together**:
By combining statements (1) and (2), we know that a^3 is divisible by 4 and (-1)^a * (-1)^(a+1) is equal to -1.
However, even with this information, we still cannot determine if f(a+1) - f(a) is divisible by 4.
For example, let's consider two cases:
Case 1: a = 2
In this case, a^3 = 2^3 = 8, which is divisible by 4. Also, (-1)^a * (-1)^(a+1) = (-1)^2 * (-1)^3 = 1 * (-1) = -1.
If we substitute a = 2 into the expression (a+1)^3 - a^3, we get:
(2+1)^3 - 2^3 = 3^3 - 8 = 27 - 8 = 19, which is not divisible by 4.
Case 2: a
In order to determine if f(a+1) - f(a) is divisible by 4, we need to understand the function f(x) = x * x^2.
Let's substitute a into the function:
f(a) = a * a^2 = a^3
So, statement (1) tells us that a^3 is divisible by 4.
To determine if f(a+1) - f(a) is divisible by 4, we can simplify the expression:
f(a+1) - f(a) = (a+1) * (a+1)^2 - a * a^2
= (a+1) * (a+1) * (a+1) - a^3
= (a+1)^3 - a^3
To check if (a+1)^3 - a^3 is divisible by 4, we can factorize it using the binomial theorem:
(a+1)^3 - a^3 = a^3 + 3*a^2 + 3*a + 1 - a^3
= 3*a^2 + 3*a + 1
Now, if we substitute a = 0 into the expression, we get:
3*(0)^2 + 3*(0) + 1 = 1
Since 1 is not divisible by 4, we can conclude that statement (1) alone is not sufficient to determine if f(a+1) - f(a) is divisible by 4.
**Statement (2) alone**: (-1)^a * (-1)^(a+1)
Statement (2) can be simplified as (-1)^a * (-1)^(a+1) = (-1)^a * (-1)^a * (-1) = (-1)^(2a) * (-1)
= 1 * (-1) = -1
This means that (-1)^a * (-1)^(a+1) is always equal to -1, regardless of the value of a. Since -1 is not divisible by 4, we can conclude that statement (2) alone is not sufficient to determine if f(a+1) - f(a) is divisible by 4.
**Statements (1) and (2) together**:
By combining statements (1) and (2), we know that a^3 is divisible by 4 and (-1)^a * (-1)^(a+1) is equal to -1.
However, even with this information, we still cannot determine if f(a+1) - f(a) is divisible by 4.
For example, let's consider two cases:
Case 1: a = 2
In this case, a^3 = 2^3 = 8, which is divisible by 4. Also, (-1)^a * (-1)^(a+1) = (-1)^2 * (-1)^3 = 1 * (-1) = -1.
If we substitute a = 2 into the expression (a+1)^3 - a^3, we get:
(2+1)^3 - 2^3 = 3^3 - 8 = 27 - 8 = 19, which is not divisible by 4.
Case 2: a
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If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an integer > 0(1) f(a) is divisible by 4(2) (-1)a < (-1)a+1a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'B'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an integer > 0(1) f(a) is divisible by 4(2) (-1)a < (-1)a+1a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an integer > 0(1) f(a) is divisible by 4(2) (-1)a < (-1)a+1a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'B'. Can you explain this answer?.
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