Question

What is the remainder of 3^29 divided by 4?

Answer 1

Since 29 is an odd number,
the remainder happens to be 3

Explanation:

`3^29/4`
when 3^0 =1 is divided by 4, the remainder is 1
when 3^1 =3 is divided by 4, the remainder is 3
when 3^2 =9 is divided by 4, the remainder is 1
when 3^3 =27 is divided by 4, the remainder is 3
ie
all the even powers of 3 has remainder 1
all the odd powers of 3 has remainder 3

Since 29 is an odd number,
the remainder happens to be 3

Answer 2

3

Explanation:

If you look at the pattern of `3^x/4` you see the following:

`3^1/4=.75`

`3^2/4=2.25`

`3^3/4=6.75`

`3^4/4=20.25`

`3^5/4=60.75`

`3^6/4=182.25`

etc.

You could make a conjecture that if the power is even, then the decimal part of the answer is equivalent to `1/4` or stated differently, the remainder is `1`. If the power is odd, then the decimal part of the answer is equivalent to `3/4` or stated differently, the remainder is `3`. Therefore, `3^29/4=(SomeGiantNumber).75`, so the remainder is `3`.