Find the units digit of ## 3^{100} ## by the use of Fermat's theorem

In summary, Fermat's theorem states that if p is a prime number and a is any positive integer, then a^p - a is divisible by p. This theorem can be used to find the units digit of a number raised to a power by reducing the exponent to a smaller number using the remainder when divided by the prime number. The use of a prime number is significant because it ensures the reduction of the exponent. To find the units digit, the remainder is used as the new exponent and the resulting number will have the same units digit as the original number raised to the original exponent. However, Fermat's theorem can only be applied to numbers that are relatively prime to the prime number being used.
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Homework Statement
Find the units digit of ## 3^{100} ## by the use of Fermat's theorem.
Relevant Equations
None.
Consider modulo ## 10 ##.
Then ## 10=5\cdot 2 ##.
Applying the Fermat's theorem produces: ## 3^{4}\equiv 1\pmod {5} ##.
This means ## (3^{4})^{25}=3^{100}\equiv 1\pmod {5} ##.
Observe that ## 3\equiv 1\pmod {2}\implies 3^{100}\equiv 1\pmod {2} ##.
Now we have ## 5\mid (3^{100}-1) ## and ## 2\mid (3^{100}-1) ##.
Thus ## (5\cdot 2)\mid (3^{100}-1)\implies 3^{100}\equiv 1\pmod {10} ##.
Therefore, the units digit of ## 3^{100} ## is ## 1 ##.
 
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Well written!
 
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FAQ: Find the units digit of ## 3^{100} ## by the use of Fermat's theorem

What is Fermat's theorem?

Fermat's theorem, also known as Fermat's Little Theorem, is a mathematical theorem that states that if p is a prime number, then for any integer a, the number a^p - a is divisible by p.

How does Fermat's theorem help in finding the units digit of 3^100?

Fermat's theorem allows us to reduce the exponent of a number to a smaller value, making it easier to calculate. In this case, we can use Fermat's theorem to reduce the exponent 100 to 4, which is much easier to compute.

What is the units digit of 3^4?

The units digit of 3^4 is 1. This can be calculated by multiplying 3 four times, which gives us 81. The units digit of 81 is 1.

How do we use Fermat's theorem to find the units digit of 3^100?

Since 100 is not a prime number, we cannot directly apply Fermat's theorem. However, we can break down 100 into its prime factors, which are 2 and 5. This means that we can use Fermat's theorem twice, first to reduce the exponent to 4 and then to 2. This gives us (3^4)^25, which simplifies to 1^25. Therefore, the units digit of 3^100 is also 1.

Why is it important to use Fermat's theorem in finding the units digit of 3^100?

Using Fermat's theorem allows us to efficiently calculate the units digit of a large number without having to perform lengthy calculations. It also helps us understand the relationship between numbers and their powers, making it a useful tool in many mathematical problems.

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