But problem says "three consecutive integers" — must be integers. - Groen Casting

Understanding the Context

What Are Three Consecutive Integers?

Three consecutive integers are three whole numbers that follow each other in order, with no gaps and no decimals. Each integer immediately succeeds the previous one — there are exactly two integers between them.

Formally, if n is an integer, then three consecutive integers can be written as:

n, n + 1, n + 2

Key Insights

For example:

• If n = 5 → the integers are 5, 6, 7
  
• If n = –3 → the integers are –3, –2, –1
  
• If n = 0 → the integers are 0, 1, 2

These numbers are fundamental in number theory, algebra, and everyday problem solving because they represent simple, predictable patterns in discrete sets.

Key Properties of Three Consecutive Integers

1. Spacing: The difference between any two consecutive integers is always exactly 1.
   (Example: 6 – 5 = 1, 7 – 6 = 1)

Final Thoughts

2. Parity Alternation:
   

   • If n is even (e.g., 4), then the integers are: even, odd, even.
     
   • If n is odd (e.g., 7), they are: odd, even, odd.
     This alternation helps identify properties like even/odd distribution.

3. Divisibility Patterns:
   Among any three consecutive integers:
   

   • One is divisible by 2 (even)
     
   • One is divisible by 3 (every third number)
     
   • At least one is divisible by 3
     For example: 4, 5, 6 → 6 is divisible by 3; 7, 8, 9 → 9 is divisible by 3

4. Sum of the Three:
   The sum of three consecutive integers is always divisible by 3.
   Proof:
   n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)
   This confirms the sum is a multiple of 3.

Example: 6 + 7 + 8 = 21 → 21 ÷ 3 = 7 ✓

Why Are They Important?