1245 is a factor of the numbers p and q.
Which of the following will ALWAYS have 1245 as a factor?
(i) p + q
(ii) p `xx` q
(iii) `p -: q`

Options

A. only (ii)
B. only (i) and (ii)
C. only (ii) and (iii)
D. all - (i), (ii) and (iii)

Answer

To solve the problem, we need to analyze each of the three statements given that 1245 is a factor of both numbers \( p \) and \( q \).

### Step 1: Understanding the Given Information
Since 1245 is a factor of both \( p \) and \( q \), we can express \( p \) and \( q \) in terms of 1245:
- Let \( p = 1245 \times m \) for some integer \( m \)
- Let \( q = 1245 \times n \) for some integer \( n \)

### Step 2: Analyzing Statement (i): \( p + q \)
Now, we will check the first statement \( p + q \):
\[ p + q = (1245 \times m) + (1245 \times n) = 1245 \times (m + n) \]
Since \( m + n \) is an integer, \( p + q \) is also divisible by 1245. Thus, **1245 is a factor of \( p + q \)**.

### Step 3: Analyzing Statement (ii): \( p \times q \)
Next, we check the second statement \( p \times q \):
\[ p \times q = (1245 \times m) \times (1245 \times n) = 1245^2 \times (m \times n) \]
Since \( m \times n \) is an integer, \( p \times q \) is also divisible by \( 1245^2 \), which means **1245 is a factor of \( p \times q \)**.

### Step 4: Analyzing Statement (iii): \( p \div q \)
Now, we check the third statement \( p \div q \):
\[ p \div q = \frac{p}{q} = \frac{1245 \times m}{1245 \times n} = \frac{m}{n} \]
The expression \( \frac{m}{n} \) is not guaranteed to be an integer unless \( n \) divides \( m \). Therefore, **1245 is not necessarily a factor of \( p \div q \)**.

### Conclusion
From our analysis:
- Statement (i) \( p + q \) has 1245 as a factor.
- Statement (ii) \( p \times q \) has 1245 as a factor.
- Statement (iii) \( p \div q \) does not necessarily have 1245 as a factor.

Thus, the statements that will always have 1245 as a factor are (i) and (ii).

### Final Answer
The correct options are (i) and (ii).

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