Question: For each positive integer n, p(n) is defined to be the product of the digits of n. for example, p(724) = 56 since 7*2*4 = 56

Which of the following statements must be true?

1. p(10n) = p(n)
2. p(n + 1) > p(n)
3. p(2n) = 2p(n)

1. None
2. I and II only
3. I and III only
4. II and III only
5. I, II, and III

Answer:

Approach Solution (1):

Let’s go through each statement given in the numeric values:

1. p(10n) = p(n)

This is not true. For example: if n = 12, p(12) = 2 since 1 * 2 = 2. However, 10n = 10 (12) = 120 and p(120) = 0 since 1*2*0 = 0. Since p(120)p(12), p(10n) = p(n) is not a true statement.

2. p(n + 1) > p(n)

This is not true. For example: if n = 19, then p(19) = 9 since 1*9 = 9. However, n + 1 = 19 + 1 = 20 and p(20) = 0 since 2*0 = 0. Since p(20) < p(19), p(n + 1) > p(n) is not a true statement. Since neither I nor II is true, it can’t be choices B, C, D or E. so the correct choice must be A. However, let’s show III is not true.

3. p(2n) = 2p(n)

For example: if n = 15, then p(15) = 5 since 1*5 = 5 and 2p(15) = 2*5 = 10. However, 2n = 2*15 = 30 and p(30) = 0 since 3*0 = 0. Since p(30)2p(15), p(2n) = 2p(n) is not a true statement.

Correct Option: A

Approach Solution (2):

1. p(10n) = p(n)

p(10n) will always have units digit as 0
p(10n) = 0
p(n) can be any integer
Not a must be true statement

2. p(n + 1) > p(n)

If n = 2; p(n + 1) = 3 and p(n) = 2
p(n + 1) > p(n)
If n = 9; p(n + 1) = 0 and p(n) = 9
p(n + 1) < p(n)
Not a must be true statement

3. p(2n) = 2*p(n)

If n = 12; p(2n) = p(24) = 8 and 2 * p(n) = 4
Not a must be true statement

Correct Option: A

Approach Solution (3):

Let n = 23
Consider III
p(2n) = p(46) = 4*6 = 24
p(n) = 2*3 = 6
24 is not equal to two times of 6
So III is not must be true ruling out C, D and E options
Come to I, any number multiplied by ten will have 0 as one of its integers and product will be zero. Ruling out B as well leaving only A as correct choice.

Correct Option: A

“For each positive integer n, p(n) is defined to be the product of the digits of n. for example, p(724) = 56 since 7*2*4 = 56”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Suggested GMAT Problem Solving Samples

• If \(2^{98}=256L+N\) , Where L and N are Integers and \(0 \leq N \leq 4\) , What is the Value of N? GMAT Problem Solving
• What is the Value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2 GMAT Problem Solving
• If A Equals the Sum of the Even Integers from 2 to 20, Inclusive GMAT Problem Solving
• Is Square Root = Always Positive? GMAT Problem Solving
• \(\frac{15^{11}-15^{10}}{14}=?\) GMAT Problem Solving
• For How Many Values of k is 12^12 the Least Common Multiple GMAT Problem Solving
• A Contractor Estimated that his 10-Man Crew Could Complete the Construction in 110 Days if There Was no Rain. GMAT Problem Solving
• If n = \(3^8\)– \(2^8\), Which of the Following is Not a Factor of n? GMAT Problem Solving
• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours. GMAT Problem Solving
• GMAT Problem Solving - The Price of Raw Materials Has Gone up by 15%
• Greg Assembles Units of a Certain Product at a Factory. GMAT Problem Solving
• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving
• Five Letters Are To Be Placed Into Five Addressed Envelopes GMAT Problem Solving
• Find The Number Of Trailing Zeros In The Product Of (1^1) GMAT Problem Solving
• Find The Altitude Of An Equilateral Triangle Whose Side is 20 GMAT Problem Solving
• Which of the following is the value of √3√0.000064? GMAT Problem Solving
• Two Dice are Thrown Simultaneously. What is the Probability of Getting Two Numbers Whose Product is Even? GMAT Problem Solving
• The Smallest 3-Digit Positive Integer Obtained By Adding Two Positive Two-Digit Numbers GMAT Problem Solving
• Train A Leaves New York at 9am Eastern Time on Monday, Headed for Los Angeles at a Constant Rate GMAT Problem Solving
• An Express Train Traveled At An Average Speed of 100 Kilometers Per Hour GMAT Problem Solving
• If m Lies Between The Integers p and s On the Number Line Shown GMAT Problem Solving
• A Cube Of Side 7 cm Is Coloured On Pair of Opposite Faces By Red, Green and Yellow Shades GMAT Problem Solving
• If x^2 − 5x − 6 = 0, which of the following could be x ? GMAT Problem Solving
• How Many Numbers Between 1 and 1000, Inclusive Have an Odd Number of Factors? GMAT Problem Solving
• GMAT Problem Solving- Given f(x) = x/(x + 1), For What Value k Does f(f(k)) = 2/3 ?
• Which Among the Following is the Smallest 7-digit Number that is Exactly Divisible by 43? GMAT Problem Solving
• During a Trip, Francine Traveled x Percent of the Total Distance at an Average Speed of 40 Miles GMAT Problem Solving
• Which of the Following Expressions CANNOT have a Negative Value? GMAT Problem Solving
• Which of the following is greatest? GMAT Problem Solving
• The Cost Price of 20 Articles is The Same as The Selling Price of x Articles GMAT Problem Solving
• In The Figure Shown, If The Area of The Shaded Region is 3 Times The Area of The Smaller Circular Region GMAT Problem Solving
• A Regular Hexagon has a Perimeter of 30 units GMAT Problem Solving
• All the Numbers 2, 3, 4, 5, 6, 7 are Assigned to the Six Faces of a Cube, One Number to Each Face GMAT Problem Solving
• If it is true that x > -2 and x < 7, which of the following must be true? GMAT Problem Solving
• A Zookeeper Counted the Heads of the Animals in a Zoo and Found it to be 80 GMAT Problem Solving
• A Shop Stores x kg of Rice. The First Customer Buys half this Amount Plus half a kg of Rice GMAT Problem Solving
• In a Class of 120 Students Numbered 1 to 120, All Even Numbered Students Opt for Physics GMAT Problem Solving
• Machine A Produces bolts at a Uniform Rate of 120 Every 40 seconds GMAT Problem Solving
• Out of 7 Consonants and 4 Vowels, How Many Words of 3 Consonants and 2 Vowels Can be Formed? GMAT Problem Solving
• 4 Bells Toll Together at 9:00 A.M. They Toll After 7, 8, 11 and 12 Seconds GMAT Problem Solving
• 12 Marbles are Selected at Random from a Large Collection of White, Red, Green and Yellow Marbles GMAT Problem Solving
• Find the greatest number that will divide 43, 91 and 183 GMAT Problem Solving
• Of the 150 Houses in a Certain Development GMAT Problem Solving
• A man can hit a target once in 4 shots. If he fires 4 shots in succession GMAT Problem Solving
• If 75 Percent of a Class Answered the First Question on a Certain Test Correctly GMAT Problem Solving
• A Clock Strikes 4 taking 9 seconds. GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving
• Find The Value Of x GMAT Problem Solving
• GMAT Problem Solving - Which of the following expressions has the greatest values?
• GMAT Problem Solving – If @ x=x^2/2x^2-2 , What is the Units Digit of @ (@4)?