Congruent number for identity element (binary operation * is defined as a*b= a+b+2 ( mod 3), a b integers)

apple2357

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So if a binary operation * is defined as a*b= a+b+2 ( mod 3), a b integers

Are possible identity elements, 1,4,7 10 etc or must the identity element be restricted to under 3?
In other words when we use a number greater than 3 we don't get our original number out but a congruent number and do congruent numbers satisfy the identity element requirement?

Any thoughts?
 
You have to state what set the operation is defined on! Is it an operation on the integers, or on the integers mod 3 (which is the set {0, 1, 2})? If you state the problem, you get to decide that - and you need to state it explicitly.

On the other hand, I don't think it makes sense if a and b are merely integers, because a*b = a+b+2 ( mod 3) doesn't define a single integer; it defines a congruence class (all integers for which the congruence is true). The way you defined the operation implies everything is to be thought of mod 3.

And in that case, 1, 4, 7, ... are the same element!
 
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