For Each Positive Integer n, p(n) is Defined to be the Product of the Digits GMAT Problem Solving

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.

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