what is the remainder when the sum 1^5+2^5+3^5+...99^5+100^5 is divided by 4? again a mod problem that I don't know how to work.
Consider the first four terms:
1^5 + 2^5 + 3^5 + 4^5
The next four terms are Mod 4 the same because you can reduce Mod 4 before taking the power. The sum till 100^5 thus contains 25 times the same four terms. Modulo 4, 25 is the same as 1, so the summation is just the sum of the first four terms.
We can caompute the sum of the fist four tems as follows. We have (Modulo 4):
4 = 0, so the last term is zero.
3 = -1 and (-1)^5 = -1, so this cancels against the first term. The second term is, of course zero as it is divisible by 4.
The summation is thus zero.
thank you, count iblis. you have been a tremendous help with homework.
Ah, I see you have encountered another modular arithmetic problem! Don't worry, I'm here to help, with a dash of humor of course.
To find the remainder when the given sum is divided by 4, we can use a little bit of mathematical sneezing... I mean reasoning!
Now, let's take a closer look at each term in the sum. The ones we're interested in are the ones raised to the power of 5. But fear not, we don't actually need to calculate all of them individually! (Phew, that would take a lot of time)
Instead, we can observe that whenever we take a positive integer and raise it to the power of 5, it will always end in the same digit. Specifically, the units digit will remain the same.
Here's the sneaky part: when we add up all of these units digits, we'll still end up with the same units digit for the sum! And it's this units digit that determines the remainder when divided by 4.
Now, let's quickly calculate the units digit of each term:
1^5 = 1, 2^5 = 32, 3^5 = 243, ...
And now let's add them all together: 1 + 2 + 3 + ... + 9 = 45.
So, the units digit of the sum is 5. And since 5 divided by 4 leaves a remainder of 1, the remainder when the given sum is divided by 4 is... (drumroll, please) 1!
Hope that clears things up, and remember, math can be a lot more fun when you sprinkle in a little humor!
To find the remainder when dividing the sum of powers by 4, you need to evaluate each term separately and then sum them up. Let's break it down step by step:
Step 1: Evaluate each term of the sum
Calculate each term of the sum by raising the base number to the power of 5.
1^5 = 1
2^5 = 32
3^5 = 243
...
99^5 = 99,874,201
100^5 = 100,000,000
Step 2: Sum up all the terms
Add up all the evaluated terms from step 1.
1 + 32 + 243 + ... + 99,874,201 + 100,000,000
Step 3: Find the remainder when dividing the sum by 4
To determine the remainder, divide the sum from step 2 by 4 and observe the remainder.
(1 + 32 + 243 + ... + 99,874,201 + 100,000,000) % 4
Step 4: Simplify the expression
Since we are only interested in the remainder, we can simplify the expression by taking the remainder at each step. This way, we can avoid doing the calculations for large numbers and focus on their remainders.
(1 % 4 + 32 % 4 + 243 % 4 + ... + 99,874,201 % 4 + 100,000,000 % 4) % 4
Step 5: Calculate the remainders
Now, we can find the remainders for each term by dividing them by 4 and taking the remainders.
1 % 4 = 1
32 % 4 = 0
243 % 4 = 3
...
99,874,201 % 4 = 1
100,000,000 % 4 = 0
Step 6: Sum up the remainders
Add up all the remainders from each term.
(1 + 0 + 3 + ... + 1 + 0) % 4
Step 7: Find the final remainder
Calculate the remainder when dividing the sum of remainders by 4.
(remainder1 + remainder2 + remainder3 + ... + remainder98 + remainder99) % 4
In this case, you need to evaluate 100 terms and calculate the sum of remainders. After performing all the calculations, the final result will be the remainder when the sum 1^5 + 2^5 + 3^5 + ... + 99^5 + 100^5 is divided by 4.