Question
Let be a fixed positive integer. Define a relation in as follows if and only is divisible by . Show that is an equivalence relation.
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Text solutionVerified
Given that if an only divisible by
Now
Reflexive
divisible by , which is true for any integer as divisible by
Symmetric
is divisible by
is divisible by
is divisible by
is divisible by
Hence is symmetric
Transitive
Let and
is divisible by and is divisible by
is divisible by
is divisible by
Hence, is transitive.
So, an equivalence relation.
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Question Text | Let be a fixed positive integer. Define a relation in as follows if and only is divisible by . Show that is an equivalence relation. |
Answer Type | Text solution:1 |
Upvotes | 150 |