NandishSS 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)
A. None
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
Try to prove that I, II and III do NOT have to be true.
Case 1: n=1
I: p(10n) = p(n)
p(10n) = p(10*1) = p(10) = 1*0 = 0.
p(n) = p(1) = 1.
Since p(10n) ≠p(n), Statement I does not have to be true.
Eliminate B, C and E.
II: p(n+1) > p(n)
In Case 1, p(10n) < p(n).
The reason is that 10n includes a digit of 0.
To show that Statement II does not have to be true, test a case in which n+1 includes a digit of 0.
Case 2: n=9, with the result that n+1 = 10
p(n+1) = p(9+1) = p(10) = 1*0 = 0.
p(n) = p(9) = 9.
Since p(n+1) < p(n), Statement II does not have to be true.
Eliminate D.
The correct answer is
A.
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