Find the modulo 4 residue of $121 \cdot 122 \cdot 123$.
To find the modulo 4 residue of a product, I can use the fact that the residue of a product is the same as the product of the residues. So I just need to find the modulo 4 residues of 121, 122, and 123, and then multiply them together and reduce again. For example, 121 is 1 more than 120, which is a multiple of 4, so the residue of 121 is 1. Similarly, 122 is 2 more than 120, so the residue of 122 is 2. And 123 is 3 more than 120, so the residue of 123 is 3. Now I have that the modulo 4 residue of $121 \cdot 122 \cdot 123$ is the same as the modulo 4 residue of $1 \cdot 2 \cdot 3$. But this is just 6, which is 2 more than 4, so the final residue is 2. 