Find the last digit of the decimal expansion of 3^1000
Answers
Answer:1
Step-by-step explanation:
exponent ending digit
3^1 3
3^2 9
3^3 7
3^4 1
3^5 3
Here is the cyclicty of 3
We can see 1000 is divisible by 4, and that's why 3^1000
ends with 1
Hope it helps
Answer:
The last digit of the decimal expansion of 3^1000 is 1.
Step-by-step explanation:
As per the data given in the questions we have to find the last digit of the decimal expansion of 3^1000.
As per the questions it is given that number is 3^1000.
Euler's Theorem states that if gcd (a,n) = 1, then aφ(n) ≡ 1 (mod n). Here φ(n) is Euler's totient function: the number of integers in {1, 2, . . ., n-1} which are relatively prime to n. When n is a prime.
The power of a number is also called exponent, it denotes how many times we need to multiply the number. If we multiply 3 say n times, then it can be written as 3^1000.
Because (3,10)=1 we can use Fermat's Theorem and see that 3^phi(10) equivalent to 3^4 equivalent to 1 (mod 10)
Therefore, the last decimal digit of 3^1000 is 1.
Hence, the last digit of the decimal expansion of 3^1000 is 1.
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