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N is a positive integer. When N is divided by 3, the remainder is 2. When N is divided by 4, the remainder is 2. What is the remainder when N is divided by 12?

 Jun 18, 2021
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$N$ divided by 3 leaves a remainder of 2 means $N = 3k+2$ for some integer $k$.  $N$ divided by 4 leaves a remainder of 2 means $N = 4\ell+2$ for some integer $\ell$.   It follows that $3k+2=4\ell+2$, so that $3k=4\ell$.  This tells us that $\ell = 3m$ for some integer $m$.   Replacing $\ell$ by $3m$ in the equation $N = 4\ell+2$ should give you the answer.

 Jun 18, 2021

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