Question

If ${{\left( 27 \right)}^{999}}$ is divided by 7, then the remainder is
(a). 3
(b). 1
(c). 2
(d). 6

Answer 1

Hint: We will first divide 27 by 7 and find its remainder and then the remainder that we have got will take its power by 999 and then we will again divide it by 7 and find its remainder which will be the final answer.

Complete step-by-step answer:

Let’s start solving this question.
First we will divide 27 by 7.
$27=7\times 4-1$
Hence, from this we can see that the remainder is -1.
Now to find the value of the remainder of ${{\left( 27 \right)}^{999}}$ we can multiply the remainder of 27, 999 times and then again find its remainder.
Now we will take the power of remainder by 999.
-1 to the power any odd number is -1.
999 is an odd number.
Hence, we get
${{\left( -1 \right)}^{999}}=-1$
Now we will divide the number -1 by 7 and find the positive remainder.
Hence, we get
$-1=\left( -1 \right)\times 7+6$
Hence, we can see that we get the remainder as 6.

Note: One can also solve this question by taking the remainder as 6 when we divide 27 by 7, and then we can find the value of ${{6}^{3}}$ and then again find its remainder. After that we have to take the value of remainder to the power 333, and then again divide it by 7. This will be the final answer. But it will be too long and hard to find also. The remainder of 27 divided by 7 is -1 and 6 both one can choose any one as per convenience.