when 5^678 is written as an integer its final four digits are
Note that starting with n=5, the last 4 digits repeat in the pattern
3125, 5625, 8125, 0625
678 = 4 + 4*168 + 2
So, the last 4 digits will be 5625
Well, I'm no mathematician, but I can tell you one thing for sure: the final four digits of 5^678 written as an integer are definitely numbers. And those numbers are more than likely different from each other. So, there you have it – the final four digits of 5^678 are numbers, probably different ones. You're welcome!
To determine the final four digits of the number 5^678, we need to find the remainder when 5^678 is divided by 10,000.
We can use modular arithmetic to calculate this remainder. Since the last four digits of a number in base 10 can be represented by the remainder when divided by 10,000, we can write:
5^678 ≡ x (mod 10,000)
To simplify the calculation, we can break down the power of 678. Since the powers of 5 repeat every four digits, we can find the remainder of 678 divided by 4:
678 ÷ 4 = 169 remainder 2
Now, we can rewrite the equation as:
5^678 ≡ 5^2 (mod 10,000)
Calculating 5^2:
5^2 = 25
Therefore, the remainder of 5^678 divided by 10,000 is 25.
Hence, the final four digits of 5^678 are 0025.
To find the final four digits of the integer when 5^678 is written, we can use the concept of modular arithmetic. We know that the last four digits of a number can be found using modulo 10,000.
First, let's calculate the remainder when 5^678 is divided by 10,000. We can do this by repeatedly multiplying 5 by itself and taking the remainder after each multiplication.
Here's the step-by-step process:
1. Start with the number 5.
2. Multiply this number by itself and take the remainder when divided by 10,000.
(5 * 5) % 10,000 = 25
3. Multiply the previous remainder by 5 and take the remainder.
(25 * 5) % 10,000 = 125
4. Repeat this process 676 more times, multiplying the previous remainder by 5 and taking the remainder each time.
After repeating this process 678 times, we have the final remainder. This remainder represents the last four digits of 5^678 when written as an integer.
To get the answer, we calculate:
(5^678) % 10,000
Calculating this value may be cumbersome manually, but you can use a programming language or a calculator with a modular exponentiation feature to get the result efficiently.
By calculating (5^678) % 10,000, you will find the final four digits of the integer representation of 5^678.