broilerc wrote:
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?
I. \(p(10n) = p(n)\)
II. \(p(n+1) > p(n)\)
III. \(p(2n) = 2p(n)\)
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A. None
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
Let’s go through each statement given in the Roman numerals.
I. p(10n) = p(n)
This is not true. For example, if n = 12, p(12) = 2 since 1 x 2 = 2. However, 10n = 10(12) = 120 and p(120) = 0 since 1 x 2 x 0 = 0. Since p(120) ≠ p(12), p(10n) = p(n) is not a true statement.
II. p(n +1) > p(n)
This is not true. For example, if n = 19, then p(19) = 9 since 1 x 9 = 9. However, n + 1 = 19 + 1 = 20 and p(20) = 0 since 2 x 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 also not true.
III. p(2n) = 2p(n)
For example, if n = 15, then p(15) = 5 since 1 x 5 = 5 and 2p(15) = 2 x 5 = 10. However, 2n = 2 x 15 = 30 and p(30) = 0 since 3 x 0 = 0. Since p(30) ≠ 2p(15), p(2n) = 2p(n) is not a true statement.
Answer: A