Which of the following in true ?

Options

A. The cube of an odd integer of the form `8q+1`, where q is an integer.
B. The square of an odd integer of the form `8q+1`, where q is an integer.
C. The fourth power of any integer is of the form `10q+1`, where q is an integer.
D. None of these

Answer

To determine which of the given statements is true, we will analyze each option step by step.

### Step 1: Analyze Option A
**Statement:** The cube of an odd integer is of the form \(8q + 1\) where \(q\) is an integer.

1. Let's denote an odd integer as \(n = 2k + 1\) for some integer \(k\).
2. The cube of \(n\) is:
\[ n^3 = (2k + 1)^3 = 8k^3 + 12k^2 + 6k + 1 \]
3. We can factor this as:
\[ n^3 = 8k(k^2 + \frac{3}{2}k) + 1 \]
This shows that \(n^3\) can be expressed in the form \(8q + 1\) where \(q = k(k^2 + \frac{3}{2}k)\).
4. Therefore, the statement is **true**.

### Step 2: Analyze Option B
**Statement:** The square of an odd integer is of the form \(8q + 1\) where \(q\) is an integer.

1. Again, let \(n = 2k + 1\).
2. The square of \(n\) is:
\[ n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 4k(k + 1) + 1 \]
3. Since \(k(k + 1)\) is always even, we can express it as:
\[ n^2 = 8m + 1 \text{ for some integer } m \]
4. Therefore, the statement is **true**.

### Step 3: Analyze Option C
**Statement:** The fourth power of any integer is of the form \(10q + 1\) where \(q\) is an integer.

1. Let \(n\) be any integer. The fourth power of \(n\) is:
\[ n^4 \]
2. We will check a few integer values:
- If \(n = 0\), \(n^4 = 0\) (not of the form \(10q + 1\)).
- If \(n = 1\), \(n^4 = 1\) (of the form \(10q + 1\)).
- If \(n = 2\), \(n^4 = 16\) (not of the form \(10q + 1\)).
- If \(n = 3\), \(n^4 = 81\) (not of the form \(10q + 1\)).
3. Since not all integers satisfy this condition, the statement is **false**.

### Step 4: Analyze Option D
**Statement:** None of the above.

Since we found that Options A and B are true, and Option C is false, we conclude that Option D is **false**.

### Final Conclusion
The correct answer is that **Option A and Option B are true**.