We have the linear cogruence:$169x \equiv 2267 \pmod{113}$
MY try:I tried manually,but it did not help.Thank you
Solve: $169x \equiv 2267 \pmod{113}$
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elementary-number-theory
congruences
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0By obvious mental arithmetic $2267\equiv7\pmod{113}$ and $169\equiv56\pmod{113}$. Does that help? – 2017-02-10
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0It's the same as 56x=7 mod 113. – 2017-02-10
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0113 is prime so $56^{112} = 1$ so $1/56 = 56^{111} $. – 2017-02-10
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0Doh. Didn't realize 56x2=112=-1. So 56=-1/2. So 56x(-14)=7. -14=99. 56x99 = 112x48 +56=113x48-48+56=7. – 2017-02-10
1 Answers
2
$$169x\equiv 56x\equiv 2267\pmod{113}$$ Multiplying by -2 on both sides, $$ -112x\equiv x\equiv 2267(-2)\equiv -14\pmod{113}$$ $$ x\equiv 99\pmod{113}$$