What will be the remainder when 1336 is divided by 2196?

When finding the remainder of a very large power, look for the smallest power of the base that leaves remainder 1 (or -1) modulo the divisor - every further multiple of that power then repeats the same remainder, since 1 raised to any power stays 1.

1. 133 = 2197.

2. 2197 = 2196 + 1, so 133 leaves remainder 1 when divided by 2196 (133 is congruent to 1 modulo 2196).

3. 36 = 3 x 12, so 1336 = (133)12.

4. Modulo 2196, (133)12 is congruent to 112 = 1.

5. So dividing 1336 by 2196 leaves remainder 1.

This matches the binomial-expansion view: 1336 = (2196 + 1)12, and every term of that expansion except the very last one (112 = 1) carries a factor of 2196 - so only the constant term 1 survives as the remainder.

Final answer: the remainder is 1.