Yes, 1 is unique because it is considered neither prime or composite.
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By definition, 1 is not prime. A prime number has exactly two unique natural number divisors; itself and one. You can understand '1 and 1' to be two numbers, but they are not unique. The same reasoning brings us to the conclusion that 1 is not composite; a composite number can be divided without remainder by at least one integer other than itself and 1.
A prime number is a positive whole number (a positive integer) that has exactly two unique positive divisors, 1 and the number itself.
The fundamental theorem of arithmetic or the unique factorisation theorem would fail.
Every whole number is unique: no two whole numbers are the same!
Well, isn't that a happy little question! If the number 1 is not considered a composite or prime number, we simply call it a "unit." Just like every color on our palette has its own unique beauty, every number has its own special role to play in the world of mathematics.