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If A≡B(mod C) then is there any logic in saying something like A≡B? Does A≡B mean anything without the context? Here's an example problem my math professor solved during class.

In part a, he takes (5^100)^2 ≡ 13^2. To me this is implying that 5^100 ≡ 13. The fact that the mod18 is not there is very confusing to me. Could someone please explain what it means for two integers to be in the same equivalence class?

example

BobaJFET
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    Not sure what you are asking. If the modulus is understood from the context, it's not unusual to stop writing it. Is that your question? – lulu Jul 17 '21 at 22:50
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    The mod $18$ is just taken to be implied there, since it's the only base used in the question – Stephen Donovan Jul 17 '21 at 22:50
  • $a \equiv b \pmod{c}$ means that BOTH $a$ and $b$ will leave the same remainder when divided by $c$. So both of them belong to the same remainder class. When it is understood from the context, generally one may drop $\bmod c$. – Anurag A Jul 17 '21 at 22:50
  • Just as of reading this I was able to find out with a google search.. That answers my question exactly. I was just confused about how and when the modulo can be dropped. – BobaJFET Jul 17 '21 at 22:55
  • Should also stress: knowing that $a^2\equiv b^2\pmod n$ does not imply that $a\equiv b\pmod n$. You should easily be able to produce counterexamples to that. – lulu Jul 17 '21 at 22:56
  • First, $a^2\equiv b^2$ hardly ever implies $a\equiv b,$ unless you are talking modulo $2.$ But yes, there can be cases where you simply assume a modulus. It shouldn’t be done carelessly, but when the context is clear, especially when writing on a blackboard or in notes, we might elide an assumed $\pmod\cdot.$ – Thomas Andrews Jul 17 '21 at 22:56
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    @BobaJFET There is no hard and fast rule, any more than there are hard and fast rules for when to use pronouns (he,she,they,it) rather than a specific name for the item or person at hand. If there is any rule, it is “do so when it won’t cause confusion.” All shorthand is to make communication less noisy, and all shorthand risks confusion. – Thomas Andrews Jul 17 '21 at 23:04

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There is no hard and fast rule for when we choose brevity over formal correctness, any more than there are hard and fast rules for when to use pronouns (he,she,they,it) rather than a specific name for the item or person at hand. If there is any rule, it is “Do so when it won’t cause confusion and it feels redundant to repeat it.” All shorthand is to make communication less noisy, and all shorthand risks confusion.

Formally, it is wrong. But almost all of our math talk is formally wrong in some way. We make discussion less noisy and easier to understand by eliminating duplicate information.

Here, we clearly are talking about the question at hand, so there is only one possible meaning we could have for for $a\equiv b.$ Here, these lines are long enough without tacking on a redundant $\pmod{18}.$ Each answer would require more than $1$ line, or smaller handwriting.

Also, if $a^2\equiv b^2\pmod n, $ it is not true, in general, that $a\equiv b\pmod n.$

There are examples where we use notation even where it is not technically required.

For example, we assume all numbers we encounter are written in base $10$ unless a base is specified.

But in a question that requires multiple bases, you will often write $13_{10}$ rather than just $13,$ even though you technically don’t need to say it, because your reader might wonder if you forgot to specify, and meant $13_7.$

Thomas Andrews
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