If *m* and *n* are integers, 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 is/are sufficient to answer the question?

Question

If *m* and *n* are integers, 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 is/are sufficient to answer the question?

Accepted Answer

**Solution:**

1. From statement A: $\frac{m}{10} + \frac{n}{10}$ is an integer.
   - This implies $\frac{m+n}{10}$ is an integer, so $m + n = 10k$ for some integer $k$.
   - However, this alone does not guarantee that *n* is divisible by 10, since *m* could compensate.

2. From statement B: $\frac{m}{7} + \frac{n}{10}$ is an integer.
   - Let this integer be $p$, so $\frac{m}{7} + \frac{n}{10} = p$.
   - Multiply both sides by 70 (LCM of 7 and 10):
     $$10m + 7n = 70p$$
   - Since $p$ is an integer, $10m + 7n$ is divisible by 70.
   - For this to hold, *m* must be divisible by 7 and *n* must be divisible by 10.
   - Therefore, *n* is divisible by 10.

3. From statement C: *n* > *m*.
   - This inequality alone does not provide any information about divisibility.

**Conclusion:**
Only statement B alone is sufficient to conclude that *n* is divisible by 10.

**Answer:** Only B is sufficient.

Answer

Answer

Answer

Answer