$\"\"$

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Let $\"\"$ be tha statement that $\"\"$ is divisible by $\"\"$ .

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$\"\"$ is true for since $\"\"$ , which is divisible by $\"\"$ .

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Assume $\"\"$ is true, where $\"\"$ is a positive integer.

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Therefore, $\"\"$ for some integer $\"\"$ .

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Prove that $\"\"$ is also true.

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Consider $\"\"$ .

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$\"\"$ .

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Multiply each side by $\"\"$ .

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$\"\"$

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$\"\"$ .

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Since $\"\"$ is an integer, $\"\"$ is also an integer.

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Hence $\"\"$ is divisible by $\"\"$ .

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Therefore, $\"\"$ is true.

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Since $\"\"$ is true for $\"\"$ and $\"\"$ implies that $\"\"$ is also true.

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Therefore, $\"\"$ is true for $\"\"$ .

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By the mathematical induction principle, $\"\"$ is divisible by $\"\"$ for all positive integers $\"\"$ .

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$\"\"$

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By the mathematical induction principle, $\"\"$ is divisible by $\"\"$ for all positive integers $\"\"$ .

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