cloveryeah said:Homework Statement
find the last digit of 3^101 X 7^202
Homework Equations
The Attempt at a Solution
i have cal. that the last digit is 7 by...:
[147]mod10 X ([3^4]mod10)^25 X ([7^4]mod10)^25 = [7]mod10 X [1]mod10 = [7]mod10
so the last digit is 7
is it correct?
The formula for finding the last digit of a number raised to a power is to take the last digit of the base number and raise it to the remainder of the power divided by 4. If the remainder is 0, the last digit will be the last digit of the base number raised to the power of 4. If the remainder is not 0, the last digit will be the last digit of the base number raised to the remainder.
The last digit of 3^101 is 7. Applying the formula, we take the last digit of 3 (which is 3) and raise it to the remainder of 101 divided by 4 (which is 1). 3^1 is equal to 3, making the last digit of 3^101 equal to 7.
The last digit of 7^202 is 1. Applying the formula, we take the last digit of 7 (which is 7) and raise it to the remainder of 202 divided by 4 (which is 2). 7^2 is equal to 49, making the last digit of 7^202 equal to 1.
The last digit of 3^101 X 7^202 is 7. Since we are multiplying the last digits of 3^101 and 7^202, we simply need to multiply 7 (the last digit of 3^101) and 1 (the last digit of 7^202) to get a final last digit of 7.
The formula for finding the last digit of a number raised to a power is valid because of the patterns in the last digits of numbers when they are raised to powers. Every number has a repeating pattern of last digits when raised to increasing powers, and the formula takes advantage of this pattern to quickly find the last digit of a number raised to a large power.