Remainder=?

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2 Answers
Mar 7, 2018

This can be calculated in a number of ways. One way using brute force is

27^1/7 has a remainder =6 .....(1)
27^2/7=729/7 has a remainder =1 .....(2)
27^3/7=19683/7 has a remainder =6 …….. (3)
27^4/7=531441/7 has a remainder =1 ….. (4)
27^5/7=14348907/7 has a remainder =6 …..(5)
27^6/7=387420489/7 has remainder =1 …. (6)

As as per emerging pattern we observe that the remainder is =6 for an odd exponent and the remainder is =1 for an even exponent.

Given exponent is 999-> odd number. Hence, remainder =6.

Mar 7, 2018

Alternate solution

Explanation:

Given number needs to be divided by 7. Hence it can be written as

(27)^999
=>(28-1)^999

In the expansion of this series, all terms which have various powers of 28 as multiplicants will be divisible by 7. Only one term which is =(-1)^999 now needs to be tested.

We see that this term (-1)^999=-1 is not divisible by 7 and therefore, we are left with remainder =-1.
Since remainder can not be =-1, we will have to stop division process for remaining terms of expansion when the last 7 remains.

This will leave remainder as 7+(-1)=6