Prove the following statement for non-zero integers a, b, c, If a divides b and b divides c, then a

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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)

• 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