Any number, rational or irrational, can be divided by 1 and itself. A number divided by 1 does not change, and a number divided by itself is equal to 1. The definition of a rational number is a number which can be expressed as the ratio of two integers. So there is division involved, but not the kind you thought.
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A prime number
Yes, but it is a prime number and is divisible only by 1 and itself.
257 is a prime number. It is only divisible by 1 and itself.
103 is a prime number; it is only divisible by 1 and itself.
113 is a prime number. It is divisible only by 1 and itself.