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Question 13

When a natural number is divided by $$11$$, the remainder is $$4$$. When the square of this number is divided by $$11$$, the remainder is

Solution

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Let the number be $$n$$

If $$n$$ leaves a remainder of $$4$$ when divided by $$11$$, than we can write $$n$$ as:

$$n=11k+4$$ , where $$k$$ is some natural number

Now, squaring both sides

$$n^2=121k^2+16+88k$$

$$n^2=121k^2+88k+11+5$$

$$n^2=11(11k^2+8k+1)+5$$

So, we can see that now if $$n^2$$ is divided by $$11$$, the leftover $$5$$ becomes the new remainder.

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