To solve the problem step by step, we will break it down into two parts as requested.
### Part A: Find the number of integers between 1 and 1000 that are divisible by either 10, 15, or 25.
1. **Identify the sets**:
- Let \( A \) be the set of integers divisible by 10.
- Let \( B \) be the set of integers divisible by 15.
- Let \( C \) be the set of integers divisible by 25.
2. **Calculate the number of elements in each set**:
- \( N(A) \): The integers divisible by 10 between 1 and 1000 are \( 10, 20, 30, \ldots, 1000 \). The last term is \( 1000 \), and the first term is \( 10 \). The number of terms can be calculated as:
\[
N(A) = \frac{1000 - 10}{10} + 1 = 100
\]
- \( N(B) \): The integers divisible by 15 are \( 15, 30, 45, \ldots, 990 \). The last term is \( 990 \), and the first term is \( 15 \). The number of terms is:
\[
N(B) = \frac{990 - 15}{15} + 1 = 66
\]
- \( N(C) \): The integers divisible by 25 are \( 25, 50, 75, \ldots, 1000 \). The last term is \( 1000 \), and the first term is \( 25 \). The number of terms is:
\[
N(C) = \frac{1000 - 25}{25} + 1 = 40
\]
3. **Calculate the intersections**:
- \( N(A \cap B) \): The integers divisible by both 10 and 15 are those divisible by \( \text{lcm}(10, 15) = 30 \). The integers are \( 30, 60, 90, \ldots, 990 \):
\[
N(A \cap B) = \frac{990 - 30}{30} + 1 = 33
\]
- \( N(A \cap C) \): The integers divisible by both 10 and 25 are those divisible by \( \text{lcm}(10, 25) = 50 \). The integers are \( 50, 100, 150, \ldots, 1000 \):
\[
N(A \cap C) = \frac{1000 - 50}{50} + 1 = 20
\]
- \( N(B \cap C) \): The integers divisible by both 15 and 25 are those divisible by \( \text{lcm}(15, 25) = 75 \). The integers are \( 75, 150, 225, \ldots, 975 \):
\[
N(B \cap C) = \frac{975 - 75}{75} + 1 = 13
\]
- \( N(A \cap B \cap C) \): The integers divisible by 10, 15, and 25 are those divisible by \( \text{lcm}(10, 15, 25) = 150 \). The integers are \( 150, 300, 450, \ldots, 900 \):
\[
N(A \cap B \cap C) = \frac{900 - 150}{150} + 1 = 6
\]
4. **Apply the principle of inclusion-exclusion**:
\[
N(A \cup B \cup C) = N(A) + N(B) + N(C) - N(A \cap B) - N(A \cap C) - N(B \cap C) + N(A \cap B \cap C)
\]
Substituting the values:
\[
N(A \cup B \cup C) = 100 + 66 + 40 - 33 - 20 - 13 + 6 = 146
\]
### Part B: Find the number of integers between 1 and 1000 that are divisible by neither 10, 15, nor 25.
1. **Calculate the total number of integers between 1 and 1000**:
\[
\text{Total integers} = 1000
\]
2. **Calculate the number of integers divisible by either 10, 15, or 25**:
From Part A, we found that \( N(A \cup B \cup C) = 146 \).
3. **Calculate the number of integers divisible by neither**:
\[
\text{Integers not divisible by 10, 15, or 25} = \text{Total integers} - N(A \cup B \cup C) = 1000 - 146 = 854
\]
### Final Answers:
- a. The number of integers between 1 and 1000 that are divisible by either 10, 15, or 25 is **146**.
- b. The number of integers between 1 and 1000 that are divisible by neither 10, 15, nor 25 is **854**.
