For all integers a, b, c, d, determine whether each of the following is TRUE or FALSE, Justify your answer:
a)If a|c and b|c, then (ab)|c2.
b)If a|b and c|d, then (ac)|(b + d).
c)If a|b, b|c and c|d, then a|d.
d)If a|b and c|d, then (a+c)|(b+d).
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a) To determine whether (ab)|c^2 is true or false, we need to understand the properties of divisibility.
The statement (ab)|c^2 means that ab divides c^2 evenly, or in other words, c^2 is a multiple of ab.
To justify the answer, we can consider a counterexample. Let's take a = 2, b = 3, c = 6.
In this case, a|c and b|c as 2 and 3 both divide 6.
However, when we calculate (ab)|c^2, we get (2*3)|6^2 = 6|36, which is false.
In this counterexample, (ab)|c^2 is not true.
Therefore, the statement a) (If a|c and b|c, then (ab)|c^2) is FALSE.
b) The statement (ac)|(b + d) means that ac divides b+d evenly, or in other words, b+d is a multiple of ac.
To justify the answer, we can use the knowledge that if a divides b and c divides d, then ac divides bd.
In this case, if a|b and c|d, then we know that ac|bd.
Since (b + d) = b + d, (ac)|(b + d) is true based on the divisibility property.
Therefore, the statement b) (If a|b and c|d, then (ac)|(b + d)) is TRUE.
c) The statement a|d means that a divides d.
To justify the answer, we can use the transitivity property of divisibility, which states that if a|b and b|c, then a|c.
In this case, if a|b, b|c, and c|d, then a|c and c|d imply that a|d by transitivity.
Therefore, the statement c) (If a|b, b|c, and c|d, then a|d) is TRUE.
d) The statement (a+c)|(b+d) means that a+c divides b+d evenly, or in other words, b+d is a multiple of a+c.
To justify the answer, we can consider a counterexample. Let's take a = 2, b = 3, c = 4, and d = 6.
In this case, a|b as 2 divides 3, and c|d as 4 divides 6. However, when we calculate (a+c)|(b+d), we get (2+4)|(3+6) = 6|9, which is false.
In this counterexample, (a+c)|(b+d) is not true.
Therefore, the statement d) (If a|b and c|d, then (a+c)|(b+d)) is FALSE.