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A remainder in mathematics is what's left over in a division problem. In the division process, the number we want to divide up is known as the dividend, while the number we are dividing by is referred to as the divisor; the result is the quotient. Let's solve a problem based on divisibility and remainder.

Prove that the number $(n^3+2\times n)$ is divisible by 3 for a natural number $n$.

Source

Competency

Difficulty

Suggested Book

Mathematical Circles

Divisibility and Remainder

4 out of 10

Mathematical Circle

First hint

the number $n$ can give any of the following remainders 0,1 or 2 when divided by 3.

Final Hint

if $n$ has remainder 0, therefore $n^3$ and $2n$ both are divisible by 3. Hence it is proved.

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