Remainder of trick-number divided by 9

Question

How can I calculate the remainder of something like $199\cdot 741934^{1234}$ by 9?

Answer 1

Using modulo arithmetic (mod 9):

$$199 \cdot 741934^{1234} \equiv 1 \cdot 1^{1234} \equiv 1$$

So the above number yields a remainder of 1 when divided by 9.

[We can use the trick of summing up the digits to find out that $199 \equiv 1$ (mod 9). Namely, 1+9+9 = 19, which has a remainder of 1 when divided by 9.

Can do the same for 741934.]