For each of the following claims about arbitrary integers a, b, c, and d, either show that it is

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For each of the following claims about arbitrary integers a,...

For each of the following claims about arbitrary integers $a, b, \mathrm{c}$, and $d$, either show that it is true or show that it is false. (a) If $a \mid b$ and $b \mid \mathrm{c}$, then $a b \mid \mathrm{c}$. (b) If $a \mid b$ and $a \mid \mathrm{c}$, then $a \mid(b+\mathrm{c})$. (c) If $a \mid b$ and $a \mid \mathrm{c}$, then $b \mid \mathrm{c}$. (d) If $a \mid b$, then $a^2 \mid b^2$. (e) If $a \mid b$ and $c \mid d$, then $(a+\mathrm{c}) \mid(b+d)$. (f) If $a \mid b$ and $\mathrm{c} \mid d$, then $a c \mid b d$. (g) If $a \mid b$ and $a \mid \mathrm{c}$, then $a \mid b \mathrm{c}$.

For each of the following claims about arbitrary integers $a, b, \mathrm{c}$, and $d$, either show that it is true or show that it is false.
(a) If $a \mid b$ and $b \mid \mathrm{c}$, then $a b \mid \mathrm{c}$.
(b) If $a \mid b$ and $a \mid \mathrm{c}$, then $a \mid(b+\mathrm{c})$.
(c) If $a \mid b$ and $a \mid \mathrm{c}$, then $b \mid \mathrm{c}$.
(d) If $a \mid b$, then $a^2 \mid b^2$.
(e) If $a \mid b$ and $c \mid d$, then $(a+\mathrm{c}) \mid(b+d)$.
(f) If $a \mid b$ and $\mathrm{c} \mid d$, then $a c \mid b d$.
(g) If $a \mid b$ and $a \mid \mathrm{c}$, then $a \mid b \mathrm{c}$.

Key Concepts

Divisibility

Divisibility is a fundamental concept in number theory where, for two integers x and y, we say x divides y (denoted x | y) if there exists an integer k such that y = x · k. This definition underpins many further properties and theorems in arithmetic and algebra.

Transitivity of Divisibility

The transitive property states that if one integer divides a second, and the second divides a third, then the first integer divides the third. However, care must be taken when combining divisibility statements because operations on the divisors (like multiplication) may not directly preserve the property as one might expect.

Divisibility Over Sums

A key property of divisibility is that if an integer divides two numbers separately, it also divides any linear combination of those numbers. In particular, if a divides both b and c, then a divides the sum (b + c) because any expression of the form b = a·m and c = a·n can be combined to yield b + c = a·(m+n).

Divisibility Over Products

When dealing with products, divisibility can be combined in a straightforward way. For example, if a divides b and c divides d, then b and d can be expressed in terms of multiples of a and c respectively, implying that the product a·c divides the product b·d. However, one must carefully consider the structure of the multiplication to ensure all factors align properly.

Divisibility and Powers

Raising both numbers to a power preserves the divisibility relationship. If a divides b, then b can be expressed as a multiple of a, and consequently b² can be expressed as a² times another integer (the square of the multiplier), thus establishing that a² divides b².

Counterexample Method

The counterexample method is a critical analytical tool in mathematics. It involves demonstrating that a general statement is false by providing a specific example where the premises hold but the conclusion fails. This method is particularly useful in testing divisibility claims that might appear plausible but do not hold universally.

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