Select the correct choice below and, if necessary, fill in the answer box within your choice.

ANo. As a counterexample, let $a = 2$, $b = 3$, and $n = 15$. Then, $n^2 =$ [ ], which is not divisible by both $a$ and $b$.
BNo. As a counterexample, let $a = 2$, $b = 3$, and $n = 6$. Then, $n^2 =$ [ ], which is not divisible by both $a$ and $b$.
CYes. If $a \mid n$, there exists a whole number $c$ such that $ca = n$. If $b \mid n$, there exists a whole number $d$ such that $db = n$. Therefore, $(ca)(db) = n^2$. This can be rewritten as $(dc)(ab) = n^2$, which means $ab \mid n^2$, and so the statement is true.
DYes. If $a \mid n$, there exists a prime number $c$ such that $ca = n$. If $b \mid n$, there exists a prime number $d$ such that $db = n$. Therefore, $(ca)(db) = n^2$. This can be rewritten as $(dc)(ab) = n^2$, which means $ab \mid n^2$, and so the statement is true.

Question

31-10-2026
Mathematics

Answered

Select the correct choice below and, if necessary, fill in the answer box within your choice.

ANo. As a counterexample, let $a = 2$, $b = 3$, and $n = 15$. Then, $n^2 =$ [ ], which is not divisible by both $a$ and $b$.
BNo. As a counterexample, let $a = 2$, $b = 3$, and $n = 6$. Then, $n^2 =$ [ ], which is not divisible by both $a$ and $b$.
CYes. If $a \mid n$, there exists a whole number $c$ such that $ca = n$. If $b \mid n$, there exists a whole number $d$ such that $db = n$. Therefore, $(ca)(db) = n^2$. This can be rewritten as $(dc)(ab) = n^2$, which means $ab \mid n^2$, and so the statement is true.
DYes. If $a \mid n$, there exists a prime number $c$ such that $ca = n$. If $b \mid n$, there exists a prime number $d$ such that $db = n$. Therefore, $(ca)(db) = n^2$. This can be rewritten as $(dc)(ab) = n^2$, which means $ab \mid n^2$, and so the statement is true.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-10-31T12:06:06-04:00

The problem asks to verify if $a \mid n$ and $b \mid n$ implies $ab \mid n^2$.
Using the definition of divisibility, we express $n=ca$ and $n=db$.
Multiplying these gives $n^2 = (ca)(db) = (cd)(ab)$, which confirms that $ab$ divides $n^2$.
After evaluating the choices, we conclude that Option C correctly identifies the proof and confirms the statement is true.
Final Answer: $\boxed{C}$

Explanation

Analysis of Divisibility Definitions

The goal is to determine the validity of the statement: "If $a \mid n$ and $b \mid n$, then $ab \mid n^2$."
By definition, if $a \mid n$, there exists an integer $c$ such that $n = ca$. Similarly, if $b \mid n$, there exists an integer $d$ such that $n = db$. These definitions form the basis of our algebraic manipulation.

Proof of the Divisibility Statement

We can compute the square of $n$ by multiplying the two expressions for $n$:
\[n^2 = n \cdot n = (ca) \cdot (db)\]
Using the associative and commutative properties of multiplication, we can regroup the factors:
\[n^2 = (c \cdot d) \cdot (a \cdot b)\]
Since $c$ and $d$ are integers, their product $k = c \cdot d$ is also an integer. Therefore, we have:
\[n^2 = k \cdot (ab)\]
By the definition of divisibility, the existence of this integer $k$ proves that $ab \mid n^2$. Thus, the statement is mathematically sound.

Evaluation of Options

Let us examine the given options:

Option A: Claims a counterexample with $a=2, b=3, n=15$. Since $2$ does not divide $15$, the premises are not met, so this is not a valid counterexample.
Option B: Claims a counterexample with $a=2, b=3, n=6$. Here, $2 \mid 6$ and $3 \mid 6$ are true. However, $n^2 = 36$, and since $36 = 6 \times 6$, $ab=6$ indeed divides $36$. Thus, this is not a counterexample.
Option C: Provides a clear and rigorous proof by demonstrating that $n^2$ can be written as a multiple of $ab$. This aligns perfectly with our proof.
Option D: Introduces an unnecessary and incorrect restriction that the quotients $c$ and $d$ must be prime numbers, which limits the scope of the proof.

Consequently, Option C is the only correct choice.

Final Conclusion

The statement is proven to be true for any integers $a$, $b$, and $n$. Therefore, the correct choice is Option C. The final answer is $\boxed{C}$.

Examples

In logistics and supply chain management, this property is useful when calculating production batches. Suppose you have two machines with cycle times of $a$ and $b$ hours, and you need to produce a total of $n$ units where both machines are involved. Knowing that if both $a$ and $b$ divide $n$, then $ab$ must divide $n^2$, helps in planning resource allocation and ensuring that the combined throughput capacity is efficiently scaled relative to the production targets. This mathematical foundation allows for reliable predictions in batch processing cycles.