What is the least number that must be added to the product 684 × 686 so that the result becomes a perfect square?

Introduction / Context:
This problem asks you to work with products of consecutive numbers and recognize a pattern that relates such products to perfect squares. The product 684 × 686 involves two numbers that are equally spaced around 685. This structure allows you to use the identity for (n − 1)(n + 1) and connect it directly to n^2. Many number system questions exploit this identity, so understanding it gives a fast and elegant solution.

Given Data / Assumptions:

• The product considered is 684 × 686.
• We must find the least number k such that 684 × 686 + k is a perfect square.
• We are looking for the smallest non negative integer that works.
• All arithmetic is performed with integers.

Concept / Approach:
The key identity is (n − 1)(n + 1) = n^2 − 1. Here, if we take n = 685, then n − 1 is 684 and n + 1 is 686. So the product 684 × 686 is exactly 685^2 − 1. To turn this into a perfect square, we simply need to add 1, which makes the expression equal to 685^2. Recognizing this pattern avoids long multiplication and trial and error with squares.

Step-by-Step Solution:
Step 1: Observe that 684 and 686 are symmetric around 685. Step 2: Let n = 685, then 684 = n − 1 and 686 = n + 1. Step 3: Use the identity (n − 1)(n + 1) = n^2 − 1. Step 4: Substitute n = 685 into this identity to get 684 × 686 = 685^2 − 1. Step 5: We want 684 × 686 + k to be a perfect square. Step 6: From the expression above, 684 × 686 + 1 = 685^2 − 1 + 1 = 685^2. Step 7: Since 685^2 is clearly a perfect square, adding k = 1 makes the result a perfect square. Step 8: Any smaller non negative k must be 0, but 684 × 686 itself equals 685^2 − 1, which is not a perfect square. Step 9: Therefore the least number that must be added is 1.

Verification / Alternative check:
To verify, you can compute the product numerically. 684 × 686 equals 469224. Add 1 to obtain 469225. Now note that 685^2 is also 469225, which confirms that 469225 is a perfect square. If you try adding any other option, for instance 684 or 685 or 686, you will not get a number that is an exact square of an integer. Since 1 is the smallest positive adjustment that converts n^2 − 1 into n^2, it must be the least number required.

Why Other Options Are Wrong:
Adding 684, 685, or 686 produces numbers much larger than 469225 and they do not match any perfect square near 685^2. Adding 2 gives 469226, which lies between 685^2 and 686^2 but is not equal to either. Therefore none of those adjustments yield a perfect square result. Only adding 1 hits exactly 685^2, so the other options are not correct.

Common Pitfalls:
Some students attempt to expand 684 × 686 directly and then check many square values by hand, which is time consuming and error prone. Others might not notice the symmetry around 685 and miss the helpful identity (n − 1)(n + 1) = n^2 − 1. A further mistake is to assume the number to add must be large, when in fact the structure here shows that a very small adjustment works. Recognizing and using algebraic identities is a powerful way to simplify number system problems like this one.

Final Answer:
The least number that must be added is 1.