Difference between revisions of "1988 AIME Problems/Problem 9"
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Find the smallest positive integer whose [[perfect cube|cube]] ends in <math>888</math>. | Find the smallest positive integer whose [[perfect cube|cube]] ends in <math>888</math>. | ||
− | + | ==Solution 1== | |
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A little bit of checking tells us that the units digit must be 2. Now our cube must be in the form of <math>(10k + 2)^3</math>; using the [[binomial theorem]] gives us <math>1000k^3 + 600k^2 + 120k + 8</math>. Since we are looking for the tens digit, <math>\mod{100}</math> we get <math>20k + 8 \equiv 88 \pmod{100}</math>. This is true if the tens digit is either <math>4</math> or <math>9</math>. Casework: | A little bit of checking tells us that the units digit must be 2. Now our cube must be in the form of <math>(10k + 2)^3</math>; using the [[binomial theorem]] gives us <math>1000k^3 + 600k^2 + 120k + 8</math>. Since we are looking for the tens digit, <math>\mod{100}</math> we get <math>20k + 8 \equiv 88 \pmod{100}</math>. This is true if the tens digit is either <math>4</math> or <math>9</math>. Casework: | ||
*<math>4</math>: Then our cube must be in the form of <math>(100k + 42)^3 \equiv 3(100k)(42)^2 + 42^3 \equiv 200k + 88 \pmod{1000}</math>. Hence the lowest possible value for the hundreds digit is <math>4</math>, and so <math>442</math> is a valid solution. | *<math>4</math>: Then our cube must be in the form of <math>(100k + 42)^3 \equiv 3(100k)(42)^2 + 42^3 \equiv 200k + 88 \pmod{1000}</math>. Hence the lowest possible value for the hundreds digit is <math>4</math>, and so <math>442</math> is a valid solution. | ||
*<math>9</math>: Then our cube is <math>(100k + 92)^3 \equiv 3(100k)(92)^2 + 92^3 \equiv 200k + 688 \pmod{1000}</math>. The lowest possible value for the hundreds digit is <math>1</math>, and we get <math>192</math>. Hence, since <math>192 < 442</math>, the answer is <math>\fbox{192}</math> | *<math>9</math>: Then our cube is <math>(100k + 92)^3 \equiv 3(100k)(92)^2 + 92^3 \equiv 200k + 688 \pmod{1000}</math>. The lowest possible value for the hundreds digit is <math>1</math>, and we get <math>192</math>. Hence, since <math>192 < 442</math>, the answer is <math>\fbox{192}</math> | ||
− | + | ==Solution 2== | |
<math>n^3 \equiv 888 \pmod{1000} \implies n^3 \equiv 0 \pmod 8</math> and <math>n^3 \equiv 13 \pmod{125}</math>. | <math>n^3 \equiv 888 \pmod{1000} \implies n^3 \equiv 0 \pmod 8</math> and <math>n^3 \equiv 13 \pmod{125}</math>. | ||
<math>n \equiv 2 \pmod 5</math> due to the last digit of <math>n^3</math>. Let <math>n = 5a + 2</math>. By expanding, <math>125a^3 + 150a^2 + 60a + 8 \equiv 13 \pmod{125} \implies 5a^2 + 12a \equiv 1 \pmod{25}</math>. | <math>n \equiv 2 \pmod 5</math> due to the last digit of <math>n^3</math>. Let <math>n = 5a + 2</math>. By expanding, <math>125a^3 + 150a^2 + 60a + 8 \equiv 13 \pmod{125} \implies 5a^2 + 12a \equiv 1 \pmod{25}</math>. | ||
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Therefore, <math>n = 2 + 5(3+ 5(2+5a_2)) = 125a_2 + 67</math>. <math>n^3</math> must also be a multiple of <math>8</math>, so <math>n</math> must be even. <math>125a_2 + 67 \equiv 0 \pmod 2 \implies a_2 \equiv 1 \pmod 2</math>. Therefore, <math>a_2 = 2a_3 + 1</math>, where <math>a_3</math> is any non-negative integer. The number <math>n</math> has form <math>125(2a_3+1)+67 = 250a_3+192</math>. So the minimum <math>n = \boxed{192}</math>. | Therefore, <math>n = 2 + 5(3+ 5(2+5a_2)) = 125a_2 + 67</math>. <math>n^3</math> must also be a multiple of <math>8</math>, so <math>n</math> must be even. <math>125a_2 + 67 \equiv 0 \pmod 2 \implies a_2 \equiv 1 \pmod 2</math>. Therefore, <math>a_2 = 2a_3 + 1</math>, where <math>a_3</math> is any non-negative integer. The number <math>n</math> has form <math>125(2a_3+1)+67 = 250a_3+192</math>. So the minimum <math>n = \boxed{192}</math>. | ||
− | + | == Solution 3 == | |
Let <math>x^3 = 1000a + 888</math>. We factor an <math>8</math> out of the right hand side, and we note that <math>x</math> must be of the form <math>x = 2y</math>, where <math>y</math> is a positive integer. Then, this becomes <math>y^3 = 125a + 111</math>. Taking mod <math>5</math>, <math>25</math>, and <math>125</math>, we get <math>y^3 \equiv 1\pmod 5</math>, <math>y^3 \equiv 11\pmod{25}</math>, and <math>y^3 \equiv 111\pmod{125}</math>. | Let <math>x^3 = 1000a + 888</math>. We factor an <math>8</math> out of the right hand side, and we note that <math>x</math> must be of the form <math>x = 2y</math>, where <math>y</math> is a positive integer. Then, this becomes <math>y^3 = 125a + 111</math>. Taking mod <math>5</math>, <math>25</math>, and <math>125</math>, we get <math>y^3 \equiv 1\pmod 5</math>, <math>y^3 \equiv 11\pmod{25}</math>, and <math>y^3 \equiv 111\pmod{125}</math>. | ||
We can work our way up, and find that <math>y\equiv 1\pmod 5</math>, <math>y\equiv 21\pmod{25}</math>, and finally <math>y\equiv 96\pmod{125}</math>. This gives us our smallest value, <math>y = 96</math>, so <math>x = \boxed{192}</math>, as desired. - Spacesam | We can work our way up, and find that <math>y\equiv 1\pmod 5</math>, <math>y\equiv 21\pmod{25}</math>, and finally <math>y\equiv 96\pmod{125}</math>. This gives us our smallest value, <math>y = 96</math>, so <math>x = \boxed{192}</math>, as desired. - Spacesam | ||
− | + | == Solution 4 (Bash) == | |
Let this integer be <math>x.</math> Note that <cmath>x^3 \equiv 888 \pmod{1000} \implies x \equiv 0 \pmod {2}~~ \cap ~~ x \equiv 2 \pmod{5}.</cmath> We wish to find the residue of <math>x</math> mod <math>125.</math> Note that <cmath>x \equiv 2,7,12,17, \text{ or } 22 \pmod{25}</cmath> using our congruence in mod <math>5.</math> The residue that works must also satisfy <math>x^3 \equiv 13 \pmod{25}</math> from our original congruence. Noting that <math>17^3 \equiv (-8)^3 \equiv -512 \equiv 13 \pmod{25}</math> (and bashing out the other residues perhaps but they're not that hard), we find that <cmath>x \equiv 17 \pmod{25}.</cmath> Thus, <cmath>x \equiv 17,42,67,92,117 \pmod{125}.</cmath> The residue that works must also satisfy <math>x^3 \equiv 13 \pmod{125}</math> from our original congruence. It is easy to memorize that <cmath>17^3 \equiv \mathbf{4913} \equiv 38 \pmod{125}.</cmath> Also, <cmath>42^3 \equiv 42^2 \cdot 42 \equiv 1764 \cdot 42 \equiv 14 \cdot 42 \equiv 88 \pmod{125}.</cmath> Finally, <cmath>67^3 \equiv 67^2 \cdot 67 \equiv 4489 \cdot 67 \equiv (-11) \cdot 67 \equiv -737 \equiv 13 \pmod{125},</cmath> as desired. Thus, <math>x</math> must satisfy <cmath>x \equiv 0 \pmod{2}~~ \cap ~~x \equiv 67 \pmod{125} \implies x \equiv 192 \pmod{250} \implies x=\boxed{192}.</cmath> ~samrocksnature | Let this integer be <math>x.</math> Note that <cmath>x^3 \equiv 888 \pmod{1000} \implies x \equiv 0 \pmod {2}~~ \cap ~~ x \equiv 2 \pmod{5}.</cmath> We wish to find the residue of <math>x</math> mod <math>125.</math> Note that <cmath>x \equiv 2,7,12,17, \text{ or } 22 \pmod{25}</cmath> using our congruence in mod <math>5.</math> The residue that works must also satisfy <math>x^3 \equiv 13 \pmod{25}</math> from our original congruence. Noting that <math>17^3 \equiv (-8)^3 \equiv -512 \equiv 13 \pmod{25}</math> (and bashing out the other residues perhaps but they're not that hard), we find that <cmath>x \equiv 17 \pmod{25}.</cmath> Thus, <cmath>x \equiv 17,42,67,92,117 \pmod{125}.</cmath> The residue that works must also satisfy <math>x^3 \equiv 13 \pmod{125}</math> from our original congruence. It is easy to memorize that <cmath>17^3 \equiv \mathbf{4913} \equiv 38 \pmod{125}.</cmath> Also, <cmath>42^3 \equiv 42^2 \cdot 42 \equiv 1764 \cdot 42 \equiv 14 \cdot 42 \equiv 88 \pmod{125}.</cmath> Finally, <cmath>67^3 \equiv 67^2 \cdot 67 \equiv 4489 \cdot 67 \equiv (-11) \cdot 67 \equiv -737 \equiv 13 \pmod{125},</cmath> as desired. Thus, <math>x</math> must satisfy <cmath>x \equiv 0 \pmod{2}~~ \cap ~~x \equiv 67 \pmod{125} \implies x \equiv 192 \pmod{250} \implies x=\boxed{192}.</cmath> ~samrocksnature |
Revision as of 11:25, 6 December 2022
Contents
Problem
Find the smallest positive integer whose cube ends in .
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 ; using the binomial theorem gives us . Since we are looking for the tens digit, we get . This is true if the tens digit is either or . Casework:
- : Then our cube must be in the form of . Hence the lowest possible value for the hundreds digit is , and so is a valid solution.
- : Then our cube is . The lowest possible value for the hundreds digit is , and we get . Hence, since , the answer is
Solution 2
and . due to the last digit of . Let . By expanding, .
By looking at the last digit again, we see , so we let where . Plugging this in to gives . Obviously, , so we let where can be any non-negative integer.
Therefore, . must also be a multiple of , so must be even. . Therefore, , where is any non-negative integer. The number has form . So the minimum .
Solution 3
Let . We factor an out of the right hand side, and we note that must be of the form , where is a positive integer. Then, this becomes . Taking mod , , and , we get , , and .
We can work our way up, and find that , , and finally . This gives us our smallest value, , so , as desired. - Spacesam
Solution 4 (Bash)
Let this integer be Note that We wish to find the residue of mod Note that using our congruence in mod The residue that works must also satisfy from our original congruence. Noting that (and bashing out the other residues perhaps but they're not that hard), we find that Thus, The residue that works must also satisfy from our original congruence. It is easy to memorize that Also, Finally, as desired. Thus, must satisfy ~samrocksnature
solution 5
This number is in the form of , after binomial expansion, we only want . We realize that are both multiples of , we only need that , so we write
Then, we write so must be a multiple of at least, so after checking, when . So , smallest , the number is
~bluesoul
See also
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 | ||
All AIME Problems and Solutions |
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