$\"\"$

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Statement : $\"\"$ is divisible by $\"\"$ .

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The statement is True.

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Proof :

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Let $\"\"$ be the statement $\"\"$ is divisible by $\"\"$ .

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$\"\"$ is the statement that $\"\"$ is divisible by $\"\"$ , $\"\"$ is true since $\"\"$

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which is divisible by $\"\"$ .

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Let us assume that $\"\"$ , such that $\"\"$ is divisible by $\"\"$ .

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Assume $\"\"$ is true where $\"\"$ is a positive integer and we have to prove that $\"\"$ must be

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true i.e., show that $\"\"$ for some integer $\"\"$ implies that $\"\"$

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is divisible by $\"\"$ .

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$\"\"$ .

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$\"\"$ . $\"\"$

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Multiply by $\"\"$ on the L.H.S and add $\"\"$ to the R.H.S

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$\"\"$

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By the induction hypothesis we know that $\"\"$ is divisible by $\"\"$ .

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Let the equation is equal to $\"\"$ .

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$\"\"$

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$\"\"$ is divisible by $\"\"$ , so it is multiple must be as well.

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$\"\"$ is multiple of $\"\"$ and also $\"\"$ is multiple of $\"\"$ .

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Then it is multiple of $\"\"$ as well.

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By the principle of mathematical induction $\"\"$ is divisible by $\"\"$ for all positive

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integers of $\"\"$ .

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$\"\"$

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By the principle of mathematical induction $\"\"$ is divisible by $\"\"$ for all positive

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integers of $\"\"$ .