Find the remainder when ## 4444^{4444} ## is divided by ## 9 ##.

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In summary, by using modular arithmetic and the hint given, it can be shown that the remainder when ##4444^{4444}## is divided by ##9## is ##7##. This is because ##4444## is equivalent to ##7## modulo ##9## and ##4444^{4444}## can be simplified to ##4444^{3n}##, which is equivalent to ##1## modulo ##9##.
  • #1
Math100
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Homework Statement
Find the remainder when ## 4444^{4444} ## is divided by ## 9 ##.
[Hint: Observe that ## 2^{3}\equiv -1\pmod {9} ##.]
Relevant Equations
None.
Observe that ## 4444\equiv 7\pmod {9} ##.
This means ## 4444^{4444}\equiv 7^{4444}\pmod {9}\equiv 7^{4+40+400+4000}\pmod {9} ##.
Now we have
\begin{align*}
&7^{4}\equiv 7\pmod {9}\\
&7^{40}\equiv (7^{4})^{10}\pmod {9}\equiv 7^{10}\pmod {9}\equiv [(7^{4})^{2}\cdot 7^{2}]\pmod {9}\equiv 7^{4}\pmod {9}\equiv 7\pmod {9}\\
&7^{400}\equiv (7^{4})^{100}\pmod {9}\equiv 7^{100}\pmod {9}\equiv (7^{4})^{25}\pmod {9}\equiv 7^{25}\pmod {9}\equiv [(7^{4})^{6}\cdot 7]\pmod {9}\equiv (7^{6}\cdot 7)\pmod {9}\equiv 7\pmod {9}\\
&7^{4000}\equiv (7^{400})^{10}\pmod {9}\equiv 7^{10}\pmod {9}\equiv 7\pmod {9}.\\
\end{align*}
Thus ## 7^{4444}\equiv (7^{4000}\cdot 7^{400}\cdot 7^{40}\cdot 7^{4})\pmod {9}\equiv 7\pmod {9} ##.
Therefore, the remainder when ## 4444^{4444} ## is divided by ## 9 ## is ## 7 ##.
 
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  • #2
Looks good.

Maybe you could have explained a bit more. E.g. ##4444=493\cdot 9 +7## or ##7^4=49^2\equiv 4^2\equiv 7\pmod 9## and similar for ##7^{10}## and ##7^6.##
 
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  • #3
Thank you.
 
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  • #4
Alternatively, using the hint:

You have already concluded that ##4444 \equiv 7 \equiv -2 ( \mod 9)##. It follows that ##4444^3 \equiv -2^3 \equiv 1( \mod 9)## and therefore ##4444^{3n}\equiv 1 (\mod 9)##.
Since ##4444 \equiv 4\cdot 4 \equiv 1 (\mod 3)## we therefore have ##4444^{4444} \equiv 4444 \equiv 7 (\mod 9)##.
 
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FAQ: Find the remainder when ## 4444^{4444} ## is divided by ## 9 ##.

1. What is the significance of finding the remainder when ## 4444^{4444} ## is divided by ## 9 ##?

Finding the remainder when a number is divided by another number is a common mathematical operation that can help us understand the properties and patterns of numbers. In this specific case, finding the remainder of ## 4444^{4444} ## when divided by ## 9 ## can provide insights into the divisibility rules of 9 and the properties of exponents.

2. How do you calculate the remainder when ## 4444^{4444} ## is divided by ## 9 ##?

The remainder when a number is divided by another number can be calculated using the modulus operator (%). In this case, we can use a calculator or a programming language to calculate the remainder of ## 4444^{4444} ## when divided by ## 9 ##, which is 7.

3. What is the pattern of the remainders when powers of 4 are divided by 9?

When powers of 4 are divided by 9, the remainders follow a pattern of 4, 7, 1, and 4 repeating. This can be observed by calculating the remainders of ## 4^{1}, 4^{2}, 4^{3}, 4^{4},... ## and so on. This pattern can also be extended to larger powers of 4, such as ## 4^{4444} ##, where the remainder is also 7.

4. How does the remainder change when the base number is changed?

The remainder when a number is divided by another number can change depending on the base number. In this case, we are dividing ## 4444^{4444} ## by 9, but if we were to divide it by a different number, the remainder would also change. For example, if we divide ## 4444^{4444} ## by 7, the remainder would be 1.

5. Can the remainder when ## 4444^{4444} ## is divided by ## 9 ## be used for any practical applications?

While the remainder when ## 4444^{4444} ## is divided by ## 9 ## may not have direct practical applications, the concept of finding remainders can be used in various fields such as cryptography, computer science, and number theory. It can also help in understanding and solving more complex mathematical problems.

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