Find the remainder of the equation

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In summary, the task is to find the remainder of the equation \frac{18^2+2^{100}}{11}. Using the theorem that if a \equiv b\ (mod\ m),\ c \equiv d\ (mod\ m) \Rightarrow a + c \equiv b +d\ (mod\ m) and ac \equiv bd\ (mod\ m), we can find that the remainder of 18^2 is 5. However, since 2^{100} is a large number, we cannot find the remainder in the same way. We can use the theorem a\equiv b (\text{mod}\; m) \Rightarrow a^k\equiv b^k (\text{mod
  • #1
parsifal
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The task is to find the remainder of the equation:
[tex]\frac{18^2+2^{100}}{11}[/tex]

Now I know that if
[tex]a \equiv b\ (mod\ m),\ c \equiv d\ (mod\ m) \Rightarrow[/tex]
[tex]a + c \equiv b +d\ (mod\ m)[/tex] and [tex]ac \equiv bd\ (mod\ m)[/tex]

so

[tex]18^2 \equiv b\ (mod\ 11) \Rightarrow \frac{18^2}{11}=29.454545... \Rightarrow b=18^2-11\cdot 29=5[/tex]
and d<6 as the remainder b+d < 11.

But as 2^100 is so large, I can't find d the way I found b. How to find it, or is there some other more convenient way that doesn't involve separating 18^2 and 2^100?
 
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  • #2
have you tried using this [tex]a\equiv b (\text{mod}\; m) \Rightarrow a^k\equiv b^k (\text{mod}\; m)[/tex] to help?

The answer should be obvious after the use of this theorem
 
  • #3


There are a few different approaches you could take to find the remainder in this equation. One way would be to use the fact that the remainder of a number divided by 11 will be the same as the remainder of its digits divided by 11. So for 2^100, you could find the remainder of 100 divided by 11 (which is 1) and then find the remainder of 2^1 divided by 11 (which is 2). Then, using the properties of modular arithmetic that you mentioned, you could find the remainder of the entire equation.

Another approach would be to use the binomial theorem to expand 18^2 and 2^100, and then use the properties of modular arithmetic to simplify the terms and find the remainder. This may be a bit more time-consuming, but it would work for any values of a and c in the equation.

Overall, there are many different methods you could use to find the remainder in this equation. It may be helpful to try out a few different approaches and see which one works best for you.
 

What does "find the remainder of the equation" mean?

When solving a mathematical equation, the remainder refers to the number left over after division. Finding the remainder involves dividing the given numbers and determining the leftover amount.

How do I find the remainder of an equation?

To find the remainder of an equation, you can use the modulus operator (%). This operator calculates the remainder of two numbers. For example, if you have the equation 13 ÷ 5, the remainder would be 3.

Can there be a negative remainder?

No, the remainder of an equation is always a positive integer. If the result of the division is a negative number, the remainder will be the same as the divisor.

Why is finding the remainder important?

Finding the remainder of an equation can be helpful in various mathematical applications, such as simplifying fractions, finding common factors, and determining if a number is divisible by another.

Are there any shortcuts for finding the remainder?

Yes, there are certain rules and properties that can help you find the remainder of an equation quickly. For example, if the divisor is a multiple of 10, the remainder will be the same as the last digit of the dividend. Knowing these shortcuts can save time and make solving equations easier.

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