Question

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

Answer 1

Hint: To solve this problem we will first have to find the reminders of the powers of $3$ they give after being divided by $4$. Then we will have to find a trend among the different powers of $3$, so that we can conclude at any specific trend. Then depending on the trend we can guess what will be the remainder of ${3^{29}}$ after being divided by $4$. So, let us see how to solve this problem.

Complete step by step answer:
we have to find the remainder of ${3^{29}}$ divided by $4$.
First let us find the remainder of smaller powers of $3$.
So, we will use the division algorithm to find the reminders of the numbers.
We know, the division algorithm gives us,
$b = aq + r$
Where, $b = $the number which is divided
$a = $the number by which it is divided
$q = $quotient
$r = $reminder
So, first let us find the remainder of ${3^0}$ after getting divided by $4$.
Therefore, we can write,
${3^0} = 1 = 4.0 + 1$
[We know, ${3^0} = 1$]
Now, the reminder of ${3^1}$ after getting divided by $4$.
Therefore, we can write,
${3^1} = 3 = 4.0 + 3$
[We know, ${3^1} = 3$]
Now, the reminder of ${3^2}$ after getting divided by $4$.
Therefore, we can write,
${3^2} = 9 = 4.2 + 1$
[We know, ${3^2} = 9$]
Now, the reminder of ${3^3}$ after getting divided by $4$.
Therefore, we can write,
${3^3} = 27 = 4.6 + 3$
[We know, ${3^3} = 27$]
So, we can notice a general trend that the numbers with even powers of $3$ have reminders $1$ on dividing with $4$.
And, the numbers with odd powers of $3$ have reminders of $3$ on dividing with $4$.
Now, in the question, we are given the number, ${3^{29}}$.
Clearly, $29$ is an odd number.
Therefore, by observing the trend we can say that, ${3^{29}}$ will have a remainder of $3$ after dividing it with $4$.

Note:
If we look at the pattern of $\dfrac{{{3^x}}}{4}$, we see the following:
$\dfrac{{{3^1}}}{4} = 0.75$
$\dfrac{{{3^2}}}{4} = 2.25$
$\dfrac{{{3^3}}}{4} = 6.75$
$\dfrac{{{3^4}}}{4} = 20.25$
$\dfrac{{{3^5}}}{4} = 60.75$