Which among the following can be expressed as (10p + q) (10q + p), where p and q are integers ?

Options

A. 1456
B. 1205
C. 1729
D. 1500

Answer

To solve the problem of determining which expression can be represented as \((10p + q)(10q + p)\), where \(p\) and \(q\) are integers, we will follow these steps:

### Step 1: Expand the expression
We start with the expression:
\[ (10p + q)(10q + p) \]
Using the distributive property (also known as the FOIL method for binomials), we can expand this:
\[ = 10p \cdot 10q + 10p \cdot p + q \cdot 10q + q \cdot p \]
This simplifies to:
\[ = 100pq + 10p^2 + 10q^2 + pq \]

### Step 2: Combine like terms
Now, we combine the like terms in the expression:
\[ = 10p^2 + 10q^2 + 101pq \]

### Step 3: Factor out common terms
Next, we can factor out a 10 from the first two terms:
\[ = 10(p^2 + q^2) + 101pq \]

### Step 4: Recognize a perfect square
Notice that \(p^2 + q^2\) can be expressed in terms of \((p + q)^2\):
\[ p^2 + q^2 = (p + q)^2 - 2pq \]
Thus, we can rewrite our expression as:
\[ = 10((p + q)^2 - 2pq) + 101pq \]
This simplifies to:
\[ = 10(p + q)^2 + (101 - 20)pq \]

### Step 5: Analyze the expression
We need to find integer values of \(p\) and \(q\) such that the entire expression can yield specific integer results.

### Step 6: Test integer values
To find integer pairs \((p, q)\) that satisfy the expression, we can try different small integers for \(p\) and \(q\):
1. Let \(p = 1\) and \(q = 7\):
\[ = 10(1 + 7)^2 + (101 - 20)(1)(7) = 10(8^2) + 81(7) = 640 + 567 = 1207 \]
2. Let \(p = 1\) and \(q = 9\):
\[ = 10(1 + 9)^2 + (101 - 20)(1)(9) = 10(10^2) + 81(9) = 1000 + 729 = 1729 \]

### Conclusion
After testing these values, we find that the expressions yield specific integers. The answer to the question is that the expression can represent integers like 1207 and 1729.