Consecutive Integers — Sum to 170 The sum of five consecutive integers is 170. Find the product of the smallest and the largest integer in this sequence.

Introduction / Context:
This is a classic consecutive-integers problem that rewards recognizing that the average of equally spaced numbers equals the middle term. With five consecutive numbers, the central integer is simply total/5. Once the sequence is identified, the endpoints multiply easily.

Given Data / Assumptions:

• Total of 5 consecutive integers: 170.
• Let the sequence be n-2, n-1, n, n+1, n+2.
• We seek (n-2) * (n+2).

Concept / Approach:
The sum equals 5n, so n = 170 / 5 = 34. The integers are 32, 33, 34, 35, 36. The smallest times the largest is 32 * 36. Alternatively, use the identity (n-2)*(n+2) = n^2 - 4 to compute quickly once n is known.

Step-by-Step Solution:
1) Compute middle n: n = 170 / 5 = 34.2) List the sequence: 32, 33, 34, 35, 36.3) Multiply extremes: 32 * 36 = (32 * 30) + (32 * 6) = 960 + 192 = 1152.4) Optional identity: n^2 - 4 = 34^2 - 4 = 1156 - 4 = 1152.

Verification / Alternative check:
Confirm the sum: 32 + 33 + 34 + 35 + 36 = 170, ensuring the sequence is correct, thus validating the product.

Why Other Options Are Wrong:
1512, 1102, 1210, and 1120 are products from incorrect endpoint choices or arithmetic mistakes (e.g., mixing numbers or using 31/37 by error).

Common Pitfalls:
Forgetting that consecutive means step 1; miscomputing the average; arithmetic slips in multiplication under time pressure.

Final Answer:
1152