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

Feb 25, 2018

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

Feb 25, 2018

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 $\frac{1}{4}$ or stated differently, the remainder is $1$. If the power is odd, then the decimal part of the answer is equivalent to $\frac{3}{4}$ or stated differently, the remainder is $3$. Therefore, ${3}^{29} / 4 = \left(S o m e G i a n t N u m b e r\right) .75$, so the remainder is $3$.