We only have one more property left to show! This final congruence property is called the *transitive property*, the statement of which goes like this:

Suppose we have the integers a, b, c, and n already defined with n > 0. If a is congruent to b mod n and b is congruent to c mod n, then it must be true that a is congruent to c mod n.

To prove the transitive property, we first suppose that a is congruent to b mod n and that b is congruent to c mod n. By definition of congruence, there exists an integer `k`

such that `(a-b) = n*k`

, and there exists another integer `m`

such that `(b-c) = n*m`

. Our task is to show that n divides (a-c), or that there exists some integer such that (a-c) is equal to n times that integer. But,

`$(a-c) = a-b+b-c = (a-b) + (b-c)$`

and so, from the congruence definitions we made above,

`$(a-b) + (b-c) = nk + nm = n(k + m)$`

We know that `k + m`

is an integer because the sum of two integers is an integer. Therefore, we have shown that n divides (a-c), which proves that a is congruent to c mod n. We will use Python to prove the transitive property of congruences through multiple checkpoints.

### Instructions

**1.**

First, create the variables `a`

, `b`

, `c`

, and `n`

and set them to their correct values in the code editor, provided that we wanted to prove the following:

If

`$15 \equiv 9\;(mod\;2)$`

and

`$9 \equiv 5\;(mod\;2),$`

then

`$15 \equiv 5\;(mod\;2).$`

**2.**

Use our variable definitions of `a`

, `b`

, `c`

, and `n`

to print whether or not the following:

`$a \equiv b\;(mod\;n)$`

and

`$b \equiv c\;(mod\;n)$`

are true.

**3.**

Finally, use our variable definitions for `a`

, `b`

, `c`

, and `n`

to print whether or not:

`$a \equiv c\;(mod\;n)$`