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I have just read that 3x congruent to 12 modulo 15 is equivalent to x congruent to 4 modulo 5. However, I thought division could only be done in modular equations if gcd(a,n)=1 but in this case, gcd(3,15)=3. Could someone please explain to me why these are equivalent and if my understanding is correct. Thanks

John
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  • Please use MathJax. Here's a tutorial. – Arturo Magidin May 19 '22 at 17:39
  • $3x\equiv 12\pmod{!15}\Rightarrow x\equiv 4\pmod{!5},$ is not division by $,3,$ (done $!\bmod 15)$ because the modulus has has changed from $,15,$ to $,5.,$ By convention, operations (like addition, multiplication and division) denote maps whose arguments and result all lie in the same number system (ring), here $,\Bbb Z_{15} = \Bbb Z\bmod 15.,$ See the linked dupes for general results on modular division (cancellation). – Bill Dubuque May 19 '22 at 17:59

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You're both right! These are special cases of the following general result: $$ aj \equiv ak \pmod n \qquad\text{if and only if}\qquad j \equiv k \pmod{\frac n{\gcd(a,n)}}. $$ Your assertion is the case where $\gcd(a,n)=1$; the example you read deals with the case $\gcd(a,n)=a$.

Greg Martin
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