2002 AMC 12B Problems/Problem 15

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Problem

How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$ ?

$\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 8$

Solution

Let $N = \overline{abcd} = 1000a + \overline{bcd}$ , such that $\frac{N}{9} = \overline{bcd}$ . Then $1000a + \overline{bcd} = 9\overline{bcd} \Longrightarrow 125a = \overline{bcd}$ . Since $100 \le \overline{bcd} < 1000$ , from $a = 1, \ldots, 7$ we have $7$ three-digit solutions, and the answer is $\mathrm{(D)}$ .

2002 AMC 12B (Problems)
Preceded by Problem 14	Followed by Problem 16
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2002 AMC 12B Problems/Problem 15

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Problem

Solution

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