Given $245^2\equiv 1\pmod{2501}$
Find $x,y$ such that $x\cdot y=2501$. (obviously not $x=1,y=2501$)
How am I suppose to approach this problem ? I don't know how to start, can't see how the given helps me to find $x,y$.
Help please, thanks !
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3$a^2-1 = (a+1)(a-1)$. – Arturo Magidin Jul 02 '22 at 15:35
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1Please see the edit to learn how to write modular equations in MathJax. – Arturo Magidin Jul 02 '22 at 15:38
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See the $,\rm \deg f = 2,$ case in this answer in the linked dupe, and see its many "Linked" questions for more examples. – Bill Dubuque Jul 02 '22 at 16:10
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Note that the factorization of $a^2-1$ is $(a-1)(a+1)$. So, \begin{align*}245^2-1^2 &\equiv 0 \mod 2501 \\ (245-1)(245+1) & \equiv 0 \mod 2501 \\ 244\cdot 246 &\equiv 0 \mod 2501.\end{align*} Note that $244\cdot 246 = 2501 \cdot 24 \implies \frac{244}{4} \cdot \frac{246}{6} = 2501 \implies \boxed{61 \cdot 41} = 2501.$
bobeyt6
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thanks , how you get $244\cdot 246 = 2501 \cdot 24$ ? (without a calculator) – Algo Jul 02 '22 at 15:46
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You could just multiply it out and divide because there seems to be no other way(a prime factorization could help but that could be used to directly solve this problem without using the modulus). – bobeyt6 Jul 02 '22 at 15:48
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Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jul 02 '22 at 16:07