Prove the following statement for non-zero integers a, b, c,

If a divides b and b divides c, then a divides c.

If an integer a divides an integer b, then we have:

b = ax for some non-zero integer x

If an integer b divides an integer c, then we have:

c = by for some non-zero integer y

Since b = ax, we substitute this into c = by for b:

c = axy

We can write this as:

c = a(xy)

If a divides b and b divides c, then a divides c.

If an integer a divides an integer b, then we have:

b = ax for some non-zero integer x

If an integer b divides an integer c, then we have:

c = by for some non-zero integer y

Since b = ax, we substitute this into c = by for b:

c = axy

We can write this as:

c = a(xy)

- Since x and y are integers, then xy is also an integer.
- Therefore, c is the product of some integer multiplied by a
- This means a divides c

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