Subjects algebra

Division Algorithm F2C84B

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1. **State the problem:** We are given the equation $$-14 = 3(-3) - 5$$ and asked whether 3 is a divisor or dividend using the Division Algorithm. 2. **Recall the Division Algorithm:** For integers $$a$$ and $$b$$ with $$b > 0$$, there exist unique integers $$q$$ (quotient) and $$r$$ (remainder) such that: $$a = bq + r$$ with $$0 \leq r < b$$. 3. **Identify terms in the equation:** Here, $$a = -14$$, $$b = 3$$, $$q = -3$$, and $$r = -5$$. 4. **Check if the equation fits the Division Algorithm:** The remainder $$r$$ must satisfy $$0 \leq r < b$$, i.e., $$0 \leq r < 3$$. 5. **Evaluate the remainder:** Given $$r = -5$$, which is less than 0, this violates the condition for the remainder in the Division Algorithm. 6. **Conclusion:** Since $$r$$ is negative, 3 cannot be the divisor in the Division Algorithm form for this equation. Instead, 3 is acting as the dividend (the number being divided), and the equation is just an expression, not a Division Algorithm representation. **Final answer:** 3 is the dividend, not the divisor, in the given equation.