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To check which common factors divide $$5{,}625$$, we evaluate it against standard divisibility tests line by line:
- **By $$2$$**: It is not divisible by $$2$$ because the final digit ($$5$$) is not an even number.
- **By $$3$$**: It is divisible by $$3$$ because the sum of its digits ($$5 + 6 + 2 + 5 = 18$$) is evenly divisible by $$3$$.
- **By $$5$$**: It is divisible by $$5$$ because its final digit is $$5$$.
- **By $$6$$**: It is not divisible by $$6$$; a number must be divisible by both $$2$$ and $$3$$ to meet this rule, and $$5{,}625$$ fails the test for $$2$$.
- **By $$10$$**: It is not divisible by $$10$$ because it does not end in $$0$$.

Applying Divisibility Tests to $$5{,}625$$

A production supervisor is auditing a batch of 5,625 components. The supervisor determines that while the batch can be divided into 3 equal groups, it cannot be divided into 6 equal groups. Based on the divisibility tests applied to this specific quantity in your lesson, what is the specific reason 5,625 is not divisible by 6?

Divisibility Audit of 5,625 Components

Based on the standard divisibility rules applied to the quantity 5,625, recall which packaging group sizes (2, 3, 5, 6, or 10) are mathematically valid for an equal distribution and provide the specific justification for why the batch fails the test for group size 6.

Packaging Efficiency Audit: Batch 5,625

We can determine the divisibility of $$4{,}962$$ by applying fundamental divisibility rules: - By $$2$$: It is divisible by $$2$$ since it ends in the even digit $$2$$. - By $$3$$: It is divisible by $$3$$ because the sum of its digits ($$4 + 9 + 6 + 2 = 21$$) is evenly divisible by $$3$$. - By $$5$$: It is not divisible by $$5$$ since the final digit is not $$0$$ or $$5$$. - By $$6$$: It is divisible by $$6$$ because it successfully passes the divisibility tests for both $$2$$ and $$3$$. - By $$10$$: It is not divisible by $$10$$ because it does not end with $$0$$.

Applying Divisibility Tests to $$4{,}962$$

Why is $$5{,}625$$ not divisible by $$2$$?

Because $$5{,}625$$ ends in $$5$$, it is not divisible by _____.

True or False: $$5{,}625$$ is divisible by $$3$$ because $$5+6+2+5=18$$, and $$18$$ is divisible by $$3$$.

Which statement about $$5{,}625$$ is true?

Match each divisor with the correct statement about $$5{,}625$$.

Why is $$5{,}625$$ not divisible by $$10$$?

Order the steps to test whether $$5{,}625$$ is divisible by $$6$$.

Question:
Is $$5{,}625$$ divisible by $$2$$, $$3$$, $$5$$, $$6$$, and $$10$$? Explain each answer using a divisibility test.

Sample answer:
$$5{,}625$$ is not divisible by $$2$$ because its last digit, $$5$$, is not even. It is divisible by $$3$$ because $$5+6+2+5=18$$, and $$18$$ is divisible by $$3$$. It is divisible by $$5$$ because it ends in $$5$$. It is not divisible by $$6$$ because it is not divisible by both $$2$$ and $$3$$. It is not divisible by $$10$$ because it does not end in $$0$$.

Key points:
- Not divisible by $$2$$: the last digit is not even.
- Divisible by $$3$$: the digit sum is $$18$$.
- Divisible by $$5$$: the last digit is $$5$$.
- Not divisible by $$6$$: it fails the test for $$2$$.
- Not divisible by $$10$$: the last digit is not $$0$$.

Rubric:
Give one point for each correct conclusion and explanation for $$2$$, $$3$$, $$5$$, $$6$$, and $$10$$.

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