If For Any Positive Integer x, d[x] Denotes its Smallest Positive GMAT Data Sufficiency

Question: If for any positive integer x, d[x] denotes its smallest positive odd divisor and D[x] denotes its largest odd divisor, is x even?

1. D[x] – d[x] = 0

2. D[3x] = 3

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Correct Answer: E

Approach Solution 1:

First of all note that the smallest positive odd divisor of any positive integer is 1.

Thus d[x] = 1 for any positive integer x.

(1) D[x] – d[x] = 0. Since d[x] = 1, then D[x] = 1 means that 1 is the largest odd divisor of x. this implies that x is not divisible by 3, 5, 7, 9, … So, x can only be 1 or 2, 4, 8, 16… (basically any power of 2). Therefore, from this statement we get that x can be 1, so odd or (2, 4, 8, …), so even. Not sufficient.

(2) D[3x] = 3. Again x can be 1, so odd, as the largest odd divisor of 3x = 3 is 3 or x can be (2, 4, 8, …), so even, as the largest odd divisor of 3 * 2 = 6 or 2 * 4 = 12 is 3. Not sufficient.

(1) + (2) From (1) and (2) we have that x can be either 1, so odd or, so even. Not sufficient

“If for any positive integer x, d[x] denotes its smallest positive odd divisor and D[x] denotes its largest odd divisor, is x even?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Suggested GMAT Data Sufficiency Questions:

• How Many of the Students in a Certain Class are Taking Both a History and a Science Course? GMAT Data Sufficiency
• The Set S of Numbers has the following Properties GMAT Data Sufficiency
• A Contractor Combined x Tons of a Gravel Mixture that Contained 10 percent Gravel G GMAT Data Sufficiency
• If Jeff has four movies, and must choose to watch either 1, 2, or 3 different movies GMAT Data Sufficiency
• What is the probability that a student randomly selected from a class of 60 students will be a male who has brown hair? GMAT Data Sufficiency
• Does The Equation y = (x – p)(x – q) Intercept The x-axis At GMAT Data Sufficiency
• S is a Set of n Consecutive Positive Integers. Is the Mean of the Set GMAT Data Sufficiency
• P and Q are Prime Numbers Less than 70. What is the Units Digit of P*Q?, GMAT Data Sufficiency
• The Cardinality of a Finite Set is the Number of Elements in the Set., GMAT Data Sufficiency
• What is the Radius of the Circle above with Center O?, GMAT Data Sufficiency
• Is the Average of a Set of 5 Distinct Positive Integers {a, b, 6, 4, 2} Greater than the Median?, GMAT Data Sufficiency
• If \(\sqrt{3+\sqrt{x-1}}=4\) , what is the value of x?, GMAT Data Sufficiency
• If There are 78 People Working at an Office, How Many of Them are Women Older than 40?, GMAT Data Sufficiency
• From a Group of M Employees, N Will be Selected, at Random, GMAT Data Sufficiency
• If L \(\neq\) 0, is \(\frac{18K}{L}\) an Integer?GMAT Data Sufficiency
• If 0 < x < 53, What is the Value of Integer x? GMAT Data Sufficiency
• If m, r, x and y are Positive, the Ratio of m to r Equal to the Ratio GMAT Data Sufficiency
• Is P – 1 Even?, GMAT Data Sufficiency
• Buster Leaves the Trailer at Noon and Walks Towards the Studio at a Constant Rate of B Miles Per Hour., GMAT Data Sufficiency
• If ab = ac is b = 2?, GMAT Data Sufficiency
• The symbol ∆ Denotes One of The Four Arithmetic Operations GMAT DATA Sufficiency, GMAT Data Sufficiency
• Is Quadrilateral ABCD a Rectangle?, GMAT Data Sufficiency
• If \(x^2\)=\(2^x\), What is the Value of x ?, GMAT Data Sufficiency
• A Pentagon With 5 Sides Of Equal Length And 5 Interior Angles Of Equal measure is Inscribed in a Circle, GMAT Data Sufficiency
• If m is a positive integer, is \(\sqrt{m}>25\) ? , GMAT Data Sufficiency
• What is x? (1) |x| < 2 (2) |x| = 3x – 2 GMAT Data Sufficiency
• In the Picture Quadrilateral ABCD is a Parallelogram and Quadrilateral DEFG is a Rectangle GMAT Data Sufficiency
• What is the Remainder when 333^222 is Divided by 7? GMAT Data Sufficiency
• Is 1/(a - b) > b - a ? GMAT Data Sufficiency
• If x is the Product of Three Consecutive Positive Integers, Which of the Following Must be True? GMAT Data Sufficiency
• What is the value of 6x^2 + 9y^2 =? GMAT Data Sufficiency
• If x and y are integers and, is x=(y/5 +2) even? GMAT Data Sufficiency
• If m is a positive integer, is √m>25? GMAT Data Sufficiency
• If there are 78 people working at an office, how many of them are women older than 40? GMAT Data Sufficiency
• If ab = ac is b = 2? GMAT Data Sufficiency
• What is the value of \(6x^2+9y^2\)?, GMAT Data Sufficiency
• If x and y are integers and \(x=\frac{y}{5}+2\) , is xy even?, GMAT Data Sufficiency
• If xy = - 6, What is the Value of xy(x + y)? GMAT Data Sufficiency
• A Drawer Contains 8 Socks, and 2 Socks are Selected at Random Without Replacement GMAT Data Sufficiency
• The Average Price of Three Stocks is $25 per Share GMAT Data Sufficiency
• Given k is a Nonzero Integer, is k>0? GMAT Data Sufficiency
• If J≠0, What Is The Value Of J? GMAT Data Sufficiency
• Paint X Costs $15 per Gallon, and Paint Y Costs $20 per Gallon GMAT Data Sufficiency
• What is the total value of Company H's stock? GMAT Data Sufficiency
• Is PQ > 1? GMAT Data Sufficiency
• If x is a positive integer, what is the value of x? GMAT Data Sufficiency
• If 53 students are enrolled in both the CS103 (Algorithms and Data Structures) And M101 GMAT Data Sufficiency
• Buster leaves the trailer at noon and walks towards the studio at a constant rate of B miles per hour GMAT Data Sufficiency
• If sq rt{3+sq rt(x-1)} = 4, what is the value of x? GMAT Data Sufficiency
• What is the sum of a, b, and c? GMAT Data Sufficiency