Question

Remainder=?

Answer 1

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.`

Answer 2

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`