To find the remainder when [tex]\(2^{100} + 3^{100} + 4^{100} + 5^{100}\)[/tex] is divided by 7, we need to calculate the remainders of each term individually when divided by 7 and then sum these remainders.
### Step-by-Step Solution:
1. Calculate [tex]\(2^{100} \mod 7\)[/tex]:
The remainder when [tex]\(2^{100}\)[/tex] is divided by 7 is [tex]\(2\)[/tex].
2. Calculate [tex]\(3^{100} \mod 7\)[/tex]:
The remainder when [tex]\(3^{100}\)[/tex] is divided by 7 is [tex]\(4\)[/tex].
3. Calculate [tex]\(4^{100} \mod 7\)[/tex]:
The remainder when [tex]\(4^{100}\)[/tex] is divided by 7 is [tex]\(4\)[/tex].
4. Calculate [tex]\(5^{100} \mod 7\)[/tex]:
The remainder when [tex]\(5^{100}\)[/tex] is divided by 7 is [tex]\(2\)[/tex].
5. Sum the remainders:
[tex]\[
2 + 4 + 4 + 2 = 12
\][/tex]
6. Calculate the remainder when this sum is divided by 7:
[tex]\[
12 \mod 7 = 5
\][/tex]
Therefore, the remainder when [tex]\(2^{100} + 3^{100} + 4^{100} + 5^{100}\)[/tex] is divided by 7 is [tex]\( \boxed{5} \)[/tex].
