High School

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Using your observations from the previous examples, determine how many 4s will be in the following equation:

\[ 2222222222222 \times 11 = \]

Number of 4s = _____

Answer :

To determine how many 4s will be in the result of multiplying 2222222222222 by 11, we approach this step-by-step using the multiplication process.

When a number is multiplied by 11, a useful pattern emerges. Each digit in the original number contributes to two digits in the result.

Let's break it down with this specific multiplication:

  1. Write out the number:

    [tex]2222222222222[/tex]

  2. Consider the multiplication by 11:

    Using the '11 rule for multiplication', you add adjacent digits. Here's how:

    • Start from the rightmost (units) digit, simply add 0 to the last digit (since there is no digit to the right):

      [tex]2[/tex] (the last digit)

    • Moving left, add each pair of adjacent digits:

      [tex]2 + 2 = 4[/tex]

      Continue this for all pairs:

      [tex]2 + 2 = 4[/tex] again

      Repeat this process for each pair in the sequence:

      [tex]2 + 2 = 4[/tex], again and again, until you reach the leftmost digit.

  3. When you reach the first digit, just add this digit as it only has one neighbor to the right:

    Add it by a non-existent 0 for the initial carry:

    [tex]2 + 0 = 2[/tex]; this provides the first digit of the sum.

The overall multiplication using this pattern for 2222222222222 by 11 would result in a sequence where most digits become 4s, except the very first and very last.

In summary, based on the pattern, all the sums result in 4 except on the edge of the sequence. It's a pattern where most intermediate sums are four. This sequence would repeat, meaning each two adjacent 2s produce a 4.

Therefore, there will be 11 digits of 4 in the product [tex]24444444444442[/tex].

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