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No, it is an adjective. There are some verbs that may seem to be modified by unbroken (e.g. Though poorly packaged, the glasses arrived unbroken)...but it refers to the noun.
Consecutive* means to follow continuously; In unbroken or logical sequence.
an unbroken part of the circumference of a circle is known as an arc
Same as contiguous - "1, 2 & 3 are contigeous numbers"
in unbroken or logical sequence. 2 and 3 are consecutive integers. 6 and 8 are consecutive even numbers.