Difference between revisions of "1988 AIME Problems/Problem 9"

Revision as of 12:00, 8 August 2022

Problem

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

Solution

Solution 1

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 $\fbox{192}$

Solution 2

$n^3 \equiv 888 \pmod{1000} \implies n^3 \equiv 0 \pmod 8$ and $n^3 \equiv 13 \pmod{125}$ . $n \equiv 2 \pmod 5$ due to the last digit of $n^3$ . Let $n = 5a + 2$ . By expanding, $125a^3 + 150a^2 + 60a + 8 \equiv 13 \pmod{125} \implies 5a^2 + 12a \equiv 1 \pmod{25}$ .

By looking at the last digit again, we see $a \equiv 3 \pmod5$ , so we let $a = 5a_1 + 3$ where $a_1 \in \mathbb{Z^+}$ . Plugging this in to $5a^2 + 12a \equiv 1 \pmod{25}$ gives $10a_1 + 6 \equiv 1 \pmod{25}$ . Obviously, $a_1 \equiv 2 \pmod 5$ , so we let $a_1 = 5a_2 + 2$ where $a_2$ can be any non-negative integer.

Therefore, $n = 2 + 5(3+ 5(2+5a_2)) = 125a_2 + 67$ . $n^3$ must also be a multiple of $8$ , so $n$ must be even. $125a_2 + 67 \equiv 0 \pmod 2 \implies a_2 \equiv 1 \pmod 2$ . Therefore, $a_2 = 2a_3 + 1$ , where $a_3$ is any non-negative integer. The number $n$ has form $125(2a_3+1)+67 = 250a_3+192$ . So the minimum $n = \boxed{192}$ .

Solution 3

Let $x^3 = 1000a + 888$ . We factor an $8$ out of the right hand side, and we note that $x$ must be of the form $x = 2y$ , where $y$ is a positive integer. Then, this becomes $y^3 = 125a + 111$ . Taking mod $5$ , $25$ , and $125$ , we get $y^3 \equiv 1\pmod 5$ , $y^3 \equiv 11\pmod{25}$ , and $y^3 \equiv 111\pmod{125}$ .

We can work our way up, and find that $y\equiv 1\pmod 5$ , $y\equiv 21\pmod{25}$ , and finally $y\equiv 96\pmod{125}$ . This gives us our smallest value, $y = 96$ , so $x = \boxed{192}$ , as desired. - Spacesam

Solution 4 (Bash)

Let this integer be $x.$ Note that $x^3 \equiv 888 \pmod{1000} \implies x \equiv 0 \pmod {2}~~ \cap ~~ x \equiv 2 \pmod{5}.$ We wish to find the residue of $x$ mod $125.$ Note that $x \equiv 2,7,12,17, \text{ or } 22 \pmod{25}$ using our congruence in mod $5.$ The residue that works must also satisfy $x^3 \equiv 13 \pmod{25}$ from our original congruence. Noting that $17^3 \equiv (-8)^3 \equiv -512 \equiv 13 \pmod{25}$ (and bashing out the other residues perhaps but they're not that hard), we find that $x \equiv 17 \pmod{25}.$ Thus, $x \equiv 17,42,67,92,117 \pmod{125}.$ The residue that works must also satisfy $x^3 \equiv 13 \pmod{125}$ from our original congruence. It is easy to memorize that $17^3 \equiv \mathbf{4913} \equiv 38 \pmod{125}.$ Also, $42^3 \equiv 42^2 \cdot 42 \equiv 1764 \cdot 42 \equiv 14 \cdot 42 \equiv 88 \pmod{125}.$ Finally, $67^3 \equiv 67^2 \cdot 67 \equiv 4489 \cdot 67 \equiv (-11) \cdot 67 \equiv -737 \equiv 13 \pmod{125},$ as desired. Thus, $x$ must satisfy $x \equiv 0 \pmod{2}~~ \cap ~~x \equiv 67 \pmod{125} \implies x \equiv 192 \pmod{250} \implies x=\boxed{192}.$ ~samrocksnature

solution 5

This number is in the form of $10k+2$ , after binomial expansion, we only want $600k^2+120k\equiv 880 \pmod{1000}$ . We realize that $600,120$ are both multiples of $8$ , we only need that $600k^2+120k \equiv 5(\mod 125 )$ , so we write $600k^2+120k=125x+5; 120k^2+24k=25x+1, 24(5k^2+k)=25x+1, 5k^2+k\equiv -1\pmod{25}$

Then, we write $5k^2+k=25m-1, 5k^2+k+1=25m$ so $k+1$ must be a multiple of $5$ at least, so $k\equiv {-1, -6, -11, -16, -21} \pmod {25}$ after checking, when $k=-6, 5k^2+k+1=175=25\cdot 7$ . So $k\equiv -6 \pmod{25}$ , smallest $k=19$ , the number is $\boxed{192}$

~bluesoul

@@ Line 29: / Line 29: @@
 This number is in the form of <math>10k+2</math>, after binomial expansion, we only want <math>600k^2+120k\equiv 880 \pmod{1000}</math>. We realize that <math>600,120</math> are both multiples of <math>8</math>, we only need that <math>600k^2+120k \equiv 5(\mod 125 )</math>, so we write <math>600k^2+120k=125x+5; 120k^2+24k=25x+1, 24(5k^2+k)=25x+1, 5k^2+k\equiv -1\pmod{25}</math>
-Then, we write <math>5k^2+k=25m-1, 5k^2+k+1=25m</math> so <math>k+1</math> must be a multiple of <math>5</math> at least, so <math>k\equiv {-1, -6, -11, -16, -21} \pmod {25}</math> after checking, when <math>k=-6, 5k^2+k+1=175=25\cdot 7</math>. So <math>k\equiv -6 \pmod{25)</math>, smallest <math>k=19</math>, the number is <math>\boxed{192}</math>
+Then, we write <math>5k^2+k=25m-1, 5k^2+k+1=25m</math> so <math>k+1</math> must be a multiple of <math>5</math> at least, so <math>k\equiv {-1, -6, -11, -16, -21} \pmod {25}</math> after checking, when <math>k=-6, 5k^2+k+1=175=25\cdot 7</math>. So <math>k\equiv -6 \pmod{25}</math>, smallest <math>k=19</math>, the number is <math>\boxed{192}</math>
 ~bluesoul

1988 AIME (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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