13) When using Vertically and crosswise to calculate 2367 x 8346 from the right, what is the result of the third step before any carry digits have been added? 


a) 62

b) 63

c) 65

d) 66

e) 67

The Vedic mathematics technique of "Vertically and Crosswise" is useful for calculating large products like \( 2367 \times 8346 \) step-by-step, breaking the process into manageable parts. To solve this problem, let's apply the method step-by-step from the right.

The "Vertically and Crosswise" method involves multiplying digits in specific columns and then summing the intermediate results.

Here's a breakdown of how the multiplication works for \( 2367 \times 8346 \):

### Step-by-Step Breakdown

1. **Identify each digit place for both numbers:**

   \( 2367 \) can be split into:
   - Thousands place: \( 2 \)
   - Hundreds place: \( 3 \)
   - Tens place: \( 6 \)
   - Units place: \( 7 \)

   \( 8346 \) can be split into:
   - Thousands place: \( 8 \)
   - Hundreds place: \( 3 \)
   - Tens place: \( 4 \)
   - Units place: \( 6 \)

2. **Start calculating the result digit-by-digit from right to left**:

   We calculate the digits in parts, combining the products and adding any carry. Let’s analyze the third step of the process. 

   To make this more understandable, we'll label the steps starting from right to left.

### The Third Step Calculation

In the Vertically and Crosswise method, we calculate the third digit starting from the right by multiplying and adding contributions from appropriate cross-multiplications:

1. **Third Step (from right):**
   
   - Multiply the units place of one number by the hundreds place of the other and vice versa.
   - Multiply the tens place of each number together.

   Specifically, for \( 2367 \times 8346 \):

   - \( 6 \times 3 = 18 \)
   - \( 7 \times 4 = 28 \)
   - \( 6 \times 6 = 36 \)

   Now, sum these intermediate products:

   \[
   18 + 28 + 36 = 82
   \]

   Since this is the third step, we haven't yet added any carry-over digits from previous steps.

### Conclusion

The result of the third step **before any carry digits have been added** is **82**. However, none of the answer choices match this value directly, indicating a possible oversight in calculating or interpreting the question.

### Rechecking Values

When using Vertically and crosswise to calculate 2367 x 8346 from the right, what is the result of the third step before any carry digits have been added?

Discover how to calculate the product of 2367 x 8346 using the Vertically and Crosswise method. This problem breaks down complex multiplication into simpler steps and provides a detailed third-step result before any carry digits are added. Suitable for students in Grades 9-12.

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