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It is given here in Theorem 3.3, that if $a \equiv b \pmod n$; then $b = a +nq, \exists q \in \mathbb{Z}$.
I feel that is more logical to state otherwise (i.e., $a = b + nq$), as is more close to euclidean division algorithm.
The article chooses the opposite notation for ease in theorem proving, but seems weird to me, although the author justifies this by symmetry property of modulus arithmetic earlier. I feel the author's syntax is not easy to use without justification (i.e., of symmetry property of congruence arithmetic) before.

However, later the author in Theorem 3.4 takes a general view of $a,b$ as congruent quantities w.r.t. the modulus.

It leads to transformation between congruence relation and equality : $a \equiv b \pmod m \iff a \pmod m = b \pmod m$.

jiten
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    They are expressing literally the exact same concept. –  Jan 18 '18 at 23:03
  • @user296602 My mindset is like $b$ is a residue or residue + k.modulus, for any integer k. But, residue 'category' dominates. – jiten Jan 18 '18 at 23:05
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    But it really isn't. $a$ and $b$ are just integers that differ by a multiple of $n$; one isn't the residue of the other. –  Jan 18 '18 at 23:06
  • Its my fault, never though like that. – jiten Jan 18 '18 at 23:07
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    Perhaps what you're thinking of is $a \mod n = b$, where $b$ is (by definition) the residue of $a$. In that case, your interpretation is a bit more natural. –  Jan 18 '18 at 23:09
  • Never was able to distinguish $a \mod n = b$ from $a \equiv b \pmod n$. – jiten Jan 18 '18 at 23:12
  • @user296602 The point is $a \mod n = b$ is only a special case of $a \equiv b \pmod n$, as the former means : $a - kn = b, \exists k \in \mathbb{Z}$. But, the congruence relation is a far bigger one; it can easily say: $b \mod n = a$, Still better, in a general sense: $a - kn = b - ln, \exists k,l \in \mathbb{Z}$, or equivalently $(a - b ) \equiv 0 \pmod n$. – jiten Jan 19 '18 at 01:43

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Logically the two statements are equivalent due to the symmetry of congruence mod $(n)$.

Therefore, we can not make a logical decision on which one is a better choice. The author has made a decision based on the flow of his proof, so we may accept his/her choice.

Mohammad Riazi-Kermani
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