Here Is a Sample Proof That Contains an Error $a \mid b$

question 109

Essay

Here is a sample proof that contains an error. Explain why the proof is not correct.
Theorem: If $a \mid b$ and $b \mid c$ , then $a \mid c$ .
Proof: Since $a \mid b , b = a k$ .
Since $b \mid c , c = b k$ .
Therefore $c = b k = ( a k ) k = a k ^ { 2 }$ .
Therefore $a \mid c$ .

---

Definitions:

Natural Experiments

Studies where the researcher takes advantage of a natural event to study its effects on subjects, without any manipulation by the researcher.

Abnormal Behaviors

Actions that deviate from the norm or societal expectations, often indicative of psychological issues or mental health disorders.

Stress

A physical, mental, or emotional factor that causes tension and may trigger bodily or psychological responses.

Generalized

A term used to describe conditions or responses that are widespread or not specific to one instance or area.