Now check which of these are divisible by 11. - Decision Point

Understanding the Context

What Does “Divisible by 11” Mean?

A number is divisible by 11 if, when divided by 11, there is no remainder—mathematically expressed as:

A number n is divisible by 11 if n ÷ 11 = k, where k is an integer.

Key Insights

But there’s an even better way to test divisibility by 11 without actual division, especially for quick mental checks: the 11 divisibility rule.

The 11 Divisibility Rule: Step-by-Step

One of the simplest and most effective ways to check if a number is divisible by 11 is the alternating sum rule. Here’s how it works:

1. Write the number from right to left (units to highest place value).
   For example, if checking 847, write it as 7 (hundreds), 4 (tens), 8 (units) → digits are 8, 4, 7 from left to right, but evaluate right to left:
   Digit places:
   
   • Position 0 (units): 7
     
   • Position 1 (tens): 4
     
   • Position 2 (hundreds): 8

Final Thoughts

2. Alternate adding and subtracting digits starting from the rightmost digit.
   Follow this pattern:
   Add: digit at odd position (right to left — starting at 0 → even indices)
   Subtract: digit at even position
   Use the sign pattern: + − + − + (starting from the right-most digit)

3. Add up the results and see if the total is divisible by 11.
   If the final sum is 0, ±11, ±22, etc., then the original number is divisible by 11.

Example: Is 143 Divisible by 11?

Let’s apply the rule:

• Number: 143 → read right to left: 3 (pos 0), 4 (pos 1), 1 (pos 2)
  
• Compute alternating sum:
  +3 (pos 0) − 4 (pos 1) + 1 (pos 2)
  = 3 − 4 + 1 = 0
  
• Since 0 is divisible by 11, 143 is divisible by 11.
  (Indeed, 143 ÷ 11 = 13)