# Show that if a, b, c, and d are integers such that a|c and b|d, then ab|cd.

## Solution

If a|c,

then there exists an integer m such that am = c,

and if b|d,

then there exists an integer n such that bn = d.

Then cd = (am)(bn) = (mn)(ab),

and thus ab|cd.

If a|c,

then there exists an integer m such that am = c,

and if b|d,

then there exists an integer n such that bn = d.

Then cd = (am)(bn) = (mn)(ab),

and thus ab|cd.

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