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756 x 322 | 243432 | longmult_3dig | longmult | [
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} | 0 | 0 | To solve the multiplication equation \( 756 \times 322 \), we will break it down into simpler steps:
### Step 1: Break down the numbers
First, let's break down each number into more manageable parts:
- \( 756 \) can be broken down as \( 700 + 50 + 6 \).
- \( 322 \) remains the same because it's already a smaller number.
### Step 2: Multiply each part of the first number by the second number
Now, multiply each component of \( 756 \) by \( 322 \):
#### Part 1: \( 700 \times 322 \)
\[ 700 \times 322 = 700 \times (300 + 20 + 2) = (700 \times 300) + (700 \times 20) + (700 \times 2) \]
\[ = 210000 + 14000 + 1400 = 225400 \]
#### Part 2: \( 50 \times 322 \)
\[ 50 \times 322 = 50 \times (300 + 20) = (50 \times 300) + (50 \times 20) \]
\[ = 15000 + 1000 = 16000 \]
#### Part 3: \( 6 \times 322 \)
\[ 6 \times 322 = 6 \times (300 + 20) = (6 \times 300) + (6 \times 20) \]
\[ = 1800 + 120 = 1920 \]
### Step 3: Add all the results together
Now, add the results from each part:
\[ 225400 + 16000 + 1920 = 243320 \]
Therefore, the solution to the multiplication equation \( 756 \times 322 \) is:
<answer>
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"Let us verify this answer: First, 756 * 322 = 243328. The calculated result does not match our manual calculation where we got 243320. Therefore, the answer is incorrect.\n<verdict>\nIncorrect\n</verdict>\n",
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"content": "<think>\n<sample>\nTo solve the multiplication equation \\( 756 \\times 322 \\), we will break it down into simpler steps:\n\n### Step 1: Break down the numbers\nFirst, let's break down each number into more manageable parts:\n- \\( 756 \\) can be broken down as \\( 700 + 50 + 6 \\).\n- \\( 322 \\) remains the same because it's already a smaller number.\n\n### Step 2: Multiply each part of the first number by the second number\nNow, multiply each component of \\( 756 \\) by \\( 322 \\):\n\n#### Part 1: \\( 700 \\times 322 \\)\n\\[ 700 \\times 322 = 700 \\times (300 + 20 + 2) = (700 \\times 300) + (700 \\times 20) + (700 \\times 2) \\]\n\\[ = 210000 + 14000 + 1400 = 225400 \\]\n\n#### Part 2: \\( 50 \\times 322 \\)\n\\[ 50 \\times 322 = 50 \\times (300 + 20) = (50 \\times 300) + (50 \\times 20) \\]\n\\[ = 15000 + 1000 = 16000 \\]\n\n#### Part 3: \\( 6 \\times 322 \\)\n\\[ 6 \\times 322 = 6 \\times (300 + 20) = (6 \\times 300) + (6 \\times 20) \\]\n\\[ = 1800 + 120 = 1920 \\]\n\n### Step 3: Add all the results together\nNow, add the results from each part:\n\\[ 225400 + 16000 + 1920 = 243320 \\]\n\nTherefore, the solution to the multiplication equation \\( 756 \\times 322 \\) is:\n<answer>\n243320\n</answer>\n\n</sample>\n<reflect>\n",
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"Let us verify this answer: First, 756 * 322 = 243328. The calculated result does not match our manual calculation. Let me recheck.\n<verification>\nThe correct answer should be:\n<sample>\nTo solve the multiplication equation \\(756 \\times 322\\), we'll use the standard long multiplication method.\n\nHere’s how i... | [
"Let us verify this answer: First, 756 * 322 = 243328. The calculated result does not match our manual calculation where we got 243320. Therefore, the answer is incorrect.\n<verdict>\nIncorrect\n</verdict>\n",
"Let us verify this answer: First, 756 * 322 = 243328. The calculated result does not match our previous... | Let us verify this answer: First, 756 * 322 = 243328. The calculated result does not match our manual calculation. Let me recheck.
<verification>
The correct answer should be:
<sample>
To solve the multiplication equation \(756 \times 322\), we'll use the standard long multiplication method.
Here’s how it works:
1. **Multiply 756 by 2**:
\[
756 \times 2 = 1512
\]
2. **Multiply 756 by 20** (which is the same as multiplying by 2 and then adding a zero at the end):
\[
756 \times 20 = 15120
\]
3. **Multiply 756 by 300** (which is the same as multiplying by 3 and then adding two zeros at the end):
\[
756 \times 300 = 226800
\]
Next, we sum these products:
\[
1512 + 15120 + 226800 = 243432
\]
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\[
756 \times 322 = 243432
\]
Therefore, the solution is:
<answer>
243432
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"content": "<think>\n<sample>\nTo solve the multiplication equation \\( 756 \\times 322 \\), we will break it down into simpler steps:\n\n### Step 1: Break down the numbers\nFirst, let's break down each number into more manageable parts:\n- \\( 756 \\) can be broken down as \\( 700 + 50 + 6 \\).\n- \\( 322 \\) remains the same because it's already a smaller number.\n\n### Step 2: Multiply each part of the first number by the second number\nNow, multiply each component of \\( 756 \\) by \\( 322 \\):\n\n#### Part 1: \\( 700 \\times 322 \\)\n\\[ 700 \\times 322 = 700 \\times (300 + 20 + 2) = (700 \\times 300) + (700 \\times 20) + (700 \\times 2) \\]\n\\[ = 210000 + 14000 + 1400 = 225400 \\]\n\n#### Part 2: \\( 50 \\times 322 \\)\n\\[ 50 \\times 322 = 50 \\times (300 + 20) = (50 \\times 300) + (50 \\times 20) \\]\n\\[ = 15000 + 1000 = 16000 \\]\n\n#### Part 3: \\( 6 \\times 322 \\)\n\\[ 6 \\times 322 = 6 \\times (300 + 20) = (6 \\times 300) + (6 \\times 20) \\]\n\\[ = 1800 + 120 = 1920 \\]\n\n### Step 3: Add all the results together\nNow, add the results from each part:\n\\[ 225400 + 16000 + 1920 = 243320 \\]\n\nTherefore, the solution to the multiplication equation \\( 756 \\times 322 \\) is:\n<answer>\n243320\n</answer>\n\n</sample>\n<reflect>\nLet us verify this answer: First, 756 * 322 = 243328. The calculated result does not match our manual calculation. Let me recheck.\n<verification>\nThe correct answer should be:\n<sample>\nTo solve the multiplication equation \\(756 \\times 322\\), we'll use the standard long multiplication method.\n\nHere’s how it works:\n\n1. **Multiply 756 by 2**:\n \\[\n 756 \\times 2 = 1512\n \\]\n\n2. **Multiply 756 by 20** (which is the same as multiplying by 2 and then adding a zero at the end):\n \\[\n 756 \\times 20 = 15120\n \\]\n\n3. **Multiply 756 by 300** (which is the same as multiplying by 3 and then adding two zeros at the end):\n \\[\n 756 \\times 300 = 226800\n \\]\n\nNext, we sum these products:\n\\[\n1512 + 15120 + 226800 = 243432\n\\]\n\nSo, the final result is:\n\\[\n756 \\times 322 = 243432\n\\]\n\nTherefore, the solution is:\n<answer>\n243432\n</answer>\n</verify>\n\n</reflect>\n</think>\n\n",
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} | 0 | 1 | To solve the multiplication equation \( 756 \times 322 \), we will break it down into simpler steps:
### Step 1: Multiply 756 by 200
\[ 756 \times 200 = 151200 \]
### Step 2: Multiply 756 by 20
\[ 756 \times 20 = 15120 \]
### Step 3: Add the results from Steps 1 and 2
\[ 151200 + 15120 = 166320 \]
Therefore, the product of \( 756 \times 322 \) is \( 166320 \).
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