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(India RMO 2005 P2, AoPS) If \(x, y\) are integers and \(17\) divides both the expressions \(x^2 - 2xy + y^2 - 5x + 7y\) and \(x^2 - 3xy + 2y^2 + x - y\), then prove that \(17\) divides \(xy - 12x + 15y\).
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- Factorize \(x^2 - 3xy + 2y^2 + x - y\) to show that
\( x \equiv y \pmod{17}, \quad \text{ or } x \equiv 2y - 1 \pmod{17} \)
holds.
- Consider the above cases seperately, and use the divisibility of the other expression by \(17\) to obtain some congruence conditions on \(y\). Using these conditions to read \(xy - 12x + 15y\) modulo \(17\).
( single ) word
\( double \) words
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Is the given equation equivalent to
[(x - p)(y - p) = p^2?]
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Is the given equation equivalent to
\($ (x - p)(y - p) = p^2?\)$
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Is the given equation equivalent to
\( (x - p)(y - p) = p^2? \)
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definition example lemma proposition theorem corollary hint walkthrough proof remark problem solution
A positive integer $p > 1$ is called prime if its only positive divisors are $1$ and $p$.
$7$ is a prime number because its only divisors are $1$ and $7$.
Every integer greater than $1$ is divisible by a prime.
Every claim is a claim.
If $p$ is a prime and $p$ divides $ab$, then $p$ divides $a$ or $p$ divides $b$.
There are infinitely many primes.
A nice corollary.
Try considering the product of all known primes plus one.
Let’s analyze the steps of the proof in detail.
Suppose there are only finitely many primes $p_1, p_2, \dots, p_n$. Consider $Q = p_1p_2\dots p_n + 1$…
A nice corollary nice.
This theorem was first proved by Euclid around 300 BC.
Prove that there are infinitely many prime numbers.
See the proof above for a constructive argument.
See this exercise.
There are infinitely many primes.
See Euclid’s Theorem.
There are infinitely many primes.
See New Theorem.
Show that the number of primes is infinite.
See Problem 1.