Let `a = b (mod n), a'= b' (mod n)` and `d, m in N. Then which of the following need not be true?

Options

A. `a +a'= b+b'(mod n)`
B. `aa'=`bb' (mod n)`
C. `a^m = b^m (mod n)`
D. `(a/d)-= (b/d) (mod n)`

Answer

To solve the problem, we need to analyze the given conditions and the options provided. Let's break it down step by step.

### Given:
1. \( a \equiv b \mod n \)
2. \( a' \equiv b' \mod n \)
3. \( d, m \in \mathbb{N} \)

### Options:
- Option A: \( a + a' \equiv b + b' \mod n \)
- Option B: \( a \cdot a' \equiv b \cdot b' \mod n \)
- Option C: \( a^m \equiv b^m \mod n \)
- Option D: \( \frac{a}{d} \equiv \frac{b}{d} \mod n \)

### Step-by-Step Solution:

**Step 1: Analyze Option A**
- From \( a \equiv b \mod n \), we can write \( a = b + kn \) for some integer \( k \).
- Similarly, from \( a' \equiv b' \mod n \), we can write \( a' = b' + ln \) for some integer \( l \).
- Now, adding these two equations:
\[ a + a' = (b + kn) + (b' + ln) = (b + b') + (k + l)n \]
- This shows that \( a + a' \equiv b + b' \mod n \).
- Therefore, **Option A is true**.

**Step 2: Analyze Option B**
- From the congruences, we have:
\[ a \equiv b \mod n \implies a = b + kn \quad \text{for some integer } k \]
\[ a' \equiv b' \mod n \implies a' = b' + ln \quad \text{for some integer } l \]
- Now, consider the product:
\[ a \cdot a' = (b + kn)(b' + ln) = bb' + bkn + bln + knln \]
- The term \( knln \) is clearly a multiple of \( n \), but the other terms \( bkn \) and \( bln \) may not necessarily be multiples of \( n \).
- Therefore, \( a \cdot a' \equiv b \cdot b' \mod n \) does not always hold.
- Thus, **Option B need not be true**.

**Step 3: Analyze Option C**
- For \( a^m \equiv b^m \mod n \):
- Since \( a \equiv b \mod n \), raising both sides to the power of \( m \) gives:
\[ a^m \equiv b^m \mod n \]
- Hence, **Option C is true**.

**Step 4: Analyze Option D**
- For \( \frac{a}{d} \equiv \frac{b}{d} \mod n \):
- If \( a \equiv b \mod n \), then dividing both sides by \( d \) (assuming \( d \) is not zero) gives:
\[ \frac{a}{d} \equiv \frac{b}{d} \mod n \]
- Thus, **Option D is true**.

### Conclusion:
The option that need not be true is **Option B**: \( a \cdot a' \equiv b \cdot b' \mod n \).