1. ## modular arith

Find the remainder of $100^{11}+1$ divided by $11$

100 is equivalent to 1 (mod 11) and 1 is equivalent to -10 (mod 11).

Thus, $1^{11}+-10=-9$. But -9 is equivalent to 2 (mod 11). Thus, the remainder is 2.

Correct?

2. Originally Posted by sfspitfire23
Find the remainder of $100^{11}+1$ divided by $11$

100 is equivalent to 1 (mod 11) and 1 is equivalent to -10 (mod 11).

Thus, $1^{11}+-10=-9$. But -9 is equivalent to 2 (mod 11). Thus, the remainder is 2.

Correct?
Why convert the 1 to -10? You create extra work for yourself. Just keep it as 1 + 1 = 2

3. Originally Posted by sfspitfire23
Find the remainder of $100^{11}+1$ divided by $11$

100 is equivalent to 1 (mod 11) and 1 is equivalent to -10 (mod 11).

Thus, $1^{11}+-10=-9$. But -9 is equivalent to 2 (mod 11). Thus, the remainder is 2.

Correct?
For similar cases you could also use Fermat's Little Theorem. Since 100 and 11 are relatively prime with 11 a prime, we have $100^{10}\equiv 1(mod\ 11)$, then $100^{11}+1\equiv 100+1(mod\ 11)$ and 100 is congruent to 1 modulo 11. So $100^{11}+1\equiv 2 (mod\ 11)$.