What is the greatest whole number less than 1000 that: (1) can be expressed as the sum of two consecutive whole numbers, and (2) can be expressed as the sum of three consecutive whole numbers, and (3) can be expressed as the sum of five consecutive whole numbers?

We are asked to find the greatest whole number less than 1000 that satisfies three conditions:

1. It can be expressed as the sum of two consecutive whole numbers.
2. It can be expressed as the sum of three consecutive whole numbers.
3. It can be expressed as the sum of five consecutive whole numbers.

### Step 1: Expressing the number in terms of sums of consecutive whole numbers

1. **Sum of two consecutive whole numbers:**

   Let the two consecutive whole numbers be \( n \) and \( n+1 \). Their sum is:
   \[
   S_2 = n + (n + 1) = 2n + 1
   \]
   So, the number must be odd.

2. **Sum of three consecutive whole numbers:**

   Let the three consecutive whole numbers be \( n-1 \), \( n \), and \( n+1 \). Their sum is:
   \[
   S_3 = (n-1) + n + (n+1) = 3n
   \]
   So, the number must be divisible by 3.

3. **Sum of five consecutive whole numbers:**

   Let the five consecutive whole numbers be \( n-2 \), \( n-1 \), \( n \), \( n+1 \), and \( n+2 \). Their sum is:
   \[
   S_5 = (n-2) + (n-1) + n + (n+1) + (n+2) = 5n
   \]
   So, the number must be divisible by 5.

### Step 2: Finding the greatest number satisfying all conditions

From the expressions above:
- The number is odd.
- The number is divisible by 3.
- The number is divisible by 5.

Since the number is divisible by 3 and 5, it must be divisible by their least common multiple (LCM), which is \( 15 \). Therefore, the number is a multiple of 15, and since it is odd, the number must be of the form \( 15k \) where \( k \) is an odd integer.

We need to find the largest such number less than 1000. So, we consider:
\[
n = 15k
\]
where \( k \) is an odd integer, and \( n < 1000 \).

### Step 3: Find the maximum value of \( k \)

We solve:
\[
15k < 1000 \implies k < \frac{1000}{15} \approx 66.67
\]
Thus, \( k \) should be the largest odd integer less than or equal to 66, which is 65.

Now, calculate the value:
\[
n = 15 \times 65 = 975
\]

### Step 4: Verification

We check if 975 meets all three conditions:
1. **Sum of two consecutive whole numbers:**
   \[
   n = 2m + 1 \text{ for some integer } m
   \]
   \( 975 = 2 \times 487 + 1 \), so it can be expressed as the sum of 487 and 488.

2. **Sum of three consecutive whole numbers:**
   \[
   n = 3m \text{ for some integer } m
   \]
   \( 975 = 3 \times 325 \), so it can be expressed as the sum of 324, 325, and 326.

3. **Sum of five consecutive whole numbers:**
   \[
   n = 5m \text{ for some integer } m
   \]
   \( 975 = 5 \times 195 \), so it can be expressed as the sum of 193, 194, 195, 196, and 197.

### Conclusion
The greatest whole number less than 1000 that satisfies all the conditions is **975**.

Would you like further details or have any other questions?

### Related Questions:
1. How do you determine if a number is the sum of two consecutive integers?
2. Can a number be the sum of any number of consecutive integers?
3. What are other numbers less than 1000 that meet only two of the three criteria?
4. How would you express 975 as the sum of four consecutive integers?
5. What is the next highest number after 975 that can be expressed as the sum of three consecutive integers?
6. Can an even number be expressed as the sum of an odd number of consecutive integers?
7. How do sums of consecutive numbers relate to triangular numbers?
8. What is the largest even number less than 1000 that meets two of the three conditions?

**Tip:** When analyzing problems involving sums of consecutive integers, consider both the arithmetic properties and the divisibility rules for the numbers involved.

Discover how to find the greatest whole number less than 1000 that can be expressed as the sum of two, three, and five consecutive whole numbers. Learn the mathematical approach using properties of odd numbers, divisibility rules, and consecutive integers. Suitable for students in Grades 7-9.

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