Question

Perform the modular arithmetic. (7 · 15) (mod 4)

Answer 1

We could do it by first writing

$7\cdot15=(4+3)\cdot(4+11)=4^2+14\cdot4+33$

so that

$7\cdot15\equiv4^2+14\cdot4+33\equiv33\pmod4$

and since $33=32+1=8\cdot4+1$, we have

$7\cdot15\equiv33\equiv1\pmod4$

# # #

Alternatively, we can just compute the product and take it mod 4:

$7\cdot15=105=104+1=26\cdot4+1$

and so we get the same result of 1.