1988 AIME Problems/Problem 9

Problem

Find the smallest positive integer whose cube ends in $888$ .

Solution

A little bit of checking tells us that the units digit must be 2. Now our cube must be in the form of $(10k + 2)^3$ ; using the binomial theorem gives us $1000k^3 + 600k^2 + 120k + 8$ . Since we are looking for the tens digit, $\mod{100}$ we get $20k + 8 \equiv 88 \pmod{100}$ . This is true if the tens digit is either $4$ or $9$ . Casework:

$4$ : Then our cube must be in the form of $(100k + 42)^3 \equiv 3(100k)(42)^2 + 42^3 \equiv 200k + 88 \pmod{1000}$ . Hence the lowest possible value for the hundreds digit is $4$ , and so $442$ is a valid solution.
$9$ : Then our cube is $(100k + 92)^3 \equiv 3(100k)(92)^2 + 92^3 \equiv 200k + 688 \pmod{1000}$ . The lowest possible value for the hundreds digit is $1$ , and we get $192$ . Hence, since $192 < 442$ , the answer is $\fbox{192}$

1988 AIME (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1988 AIME Problems/Problem 9

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