# Modulo Operation/Examples/x mod 3 = 2 and x mod 5 = 3

## Example of use of Modulo Operation

Let $x \bmod 3 = 2$ and $x \bmod 5 = 3$.

Then:

$x \bmod 15 = 8$

## Proof

 $\displaystyle x \bmod 3$ $=$ $\displaystyle 2$ $\displaystyle x \bmod 5$ $=$ $\displaystyle 3$ $\displaystyle \leadsto \ \$ $\displaystyle 5 x \bmod 15$ $=$ $\displaystyle 10$ Product Distributes over Modulo Operation $\displaystyle 3 x \bmod 15$ $=$ $\displaystyle 9$

Hence by definition of congruence by the modulo operation:

 $\displaystyle 5 x$ $\equiv$ $\displaystyle 10$ $\displaystyle \pmod {15}$ $\displaystyle 3 x$ $\equiv$ $\displaystyle 9$ $\displaystyle \pmod {15}$ $\displaystyle \leadsto \ \$ $\displaystyle 2 x$ $\equiv$ $\displaystyle 1$ $\displaystyle \pmod {15}$ Modulo Subtraction is Well-Defined: $5 x = 3 x$ $\displaystyle \leadsto \ \$ $\displaystyle 2 x$ $\equiv$ $\displaystyle 16$ $\displaystyle \pmod {15}$ Definition of Congruence Modulo Integer $\displaystyle \leadsto \ \$ $\displaystyle x$ $\equiv$ $\displaystyle 8$ $\displaystyle \pmod {15}$ Common Factor Cancelling in Congruence: Corollary $\displaystyle \leadsto \ \$ $\displaystyle x \bmod 15$ $=$ $\displaystyle 8$ Definition of Congruence by Modulo Operation

$\blacksquare$