Difference between revisions of "2001 JBMO Problems/Problem 1"
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Solve the equation <math>a^3 + b^3 + c^3 = 2001</math> in positive integers. | Solve the equation <math>a^3 + b^3 + c^3 = 2001</math> in positive integers. | ||
− | == | + | ==Solution1== |
− | Note that for all positive integers <math>n,</math> the value <math>n^3</math> is congruent to <math>-1,0,1</math> [[modulo]] <math>9.</math> Since <math>2001 \equiv 3 \pmod{9},</math> we find that <math>a,b,c \equiv 1 \pmod{9}.</math> | + | Note that for all positive integers <math>n,</math> the value <math>n^3</math> is congruent to <math>-1,0,1</math> [[modulo]] <math>9.</math> Since <math>2001 \equiv 3 \pmod{9},</math> we find that <math>a^3,b^3,c^3 \equiv 1 \pmod{9}.</math> Thus, <math>a,b,c \equiv 1 \pmod{3},</math> and the only numbers congruent to <math>1</math> modulo <math>3</math> are <math>1,4,7,10.</math> |
<br> | <br> | ||
− | + | [[WLOG]], let <math>a \ge b \ge c.</math> That means <math>a^3 \ge b^3, c^3</math> and <math>3a^3 \ge 2001.</math> Thus, <math>a^3 \ge 667,</math> so <math>a = 10.</math> | |
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+ | <br> | ||
+ | Now <math>b^3 + c^3 = 1001.</math> Since <math>b^3 \ge c^3,</math> we find that <math>2b^3 \ge 1001.</math> That means <math>b = 10</math> and <math>c = 1.</math> | ||
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+ | <br> | ||
+ | In summary, the only solutions are <math>\boxed{(10,10,1),(10,1,10),(1,10,10)}.</math> | ||
+ | |||
+ | ==Solution2== | ||
+ | You can also watch remainders modulo 7 which are also -1,0,1. The rest is almost identical as in solution 1. | ||
==See Also== | ==See Also== |
Latest revision as of 11:01, 28 April 2024
Contents
Problem
Solve the equation in positive integers.
Solution1
Note that for all positive integers the value
is congruent to
modulo
Since
we find that
Thus,
and the only numbers congruent to
modulo
are
WLOG, let That means
and
Thus,
so
Now Since
we find that
That means
and
In summary, the only solutions are
Solution2
You can also watch remainders modulo 7 which are also -1,0,1. The rest is almost identical as in solution 1.
See Also
2001 JBMO (Problems • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |