If *m* and *n* are integers, then is *n* completely divisible by 10?

Given statements:

A. The value of $\frac{m}{10} + \frac{n}{10}$ is an integer.

B. The value of $\frac{m}{7} + \frac{n}{10}$ is an integer.

C. The value of *n* is greater than *m*.

Which of the following statements are sufficient to answer the question?

Question

If *m* and *n* are integers, then is *n* completely divisible by 10?

Given statements:

A. The value of $\frac{m}{10} + \frac{n}{10}$ is an integer.

B. The value of $\frac{m}{7} + \frac{n}{10}$ is an integer.

C. The value of *n* is greater than *m*.

Which of the following statements are sufficient to answer the question?

Accepted Answer

**Solution:**

We need to determine if *n* is divisible by 10, i.e., if $n \equiv 0 \pmod{10}$.

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**Given:** *m* and *n* are integers.

**Statement A:** $\frac{m}{10} + \frac{n}{10}$ is an integer.

- This means $\frac{m+n}{10}$ is an integer.
- So, $m + n$ is divisible by 10.
- But this alone does not tell us if *n* is divisible by 10 because *m* could be any integer.

**Example:**
- If $m = 1$, $n = 9$, then $m+n=10$ divisible by 10, but $n=9$ is not divisible by 10.

Hence, **A alone is not sufficient.**

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**Statement B:** $\frac{m}{7} + \frac{n}{10}$ is an integer.

- Let $\frac{m}{7} + \frac{n}{10} = k$, where $k$ is an integer.
- Rearranged: $\frac{n}{10} = k - \frac{m}{7}$
- Multiply both sides by 70 (LCM of 7 and 10):
  $$7n = 70k - 10m$$
- Since $m$, $n$, and $k$ are integers, the right side is an integer.
- This implies $7n$ is an integer (which is obvious since $n$ is integer).

Now, check divisibility by 10:
- $7n = 70k - 10m$
- The right side is divisible by 10 (both terms are multiples of 10).
- So, $7n$ is divisible by 10.

Since 7 and 10 are coprime (no common factors other than 1), for $7n$ to be divisible by 10, *n* must be divisible by 10.

Hence, **B alone is sufficient.**

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**Statement C:** *n* is greater than *m*.

- This gives no information about divisibility.
- *n* could be any integer greater than *m*.

Hence, **C alone is not sufficient.**

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**Combining statements:**
- Since B alone is sufficient, combining with others is not necessary.

**Answer:** Only B is sufficient.

**Final choice:** (b) Only B is sufficient

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