Question

If a|-7 and a|9, then a|-63

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Jon

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To prove that if a|-7 and a|9, then a|-63, we will use the definition of divisibility.

Let's start with the first statement, a|-7. This means that a is a divisor of -7, or -7 is divisible by a. We can express this using the equation -7 = ak, where k is an integer.

Similarly, the second statement, a|9, means that a is a divisor of 9, or 9 is divisible by a. We can express this using the equation 9 = am, where m is an integer.

To prove that a|-63, we need to show that -63 is divisible by a. We can express this as -63 = al, where l is an integer.

Now, let's substitute the values of -7 and 9 from the previous equations into the equation for -63.

-7 = ak

9 = am

Multiplying the two equations together, we get:

(-7)(9) = (ak)(am)

-63 = (ak)(am)

-63 = a^2(km) (since we can multiply the exponents)

Since km is an integer (km = n, where n is an integer), we can rewrite the equation as:

-63 = a^2n

Therefore, we can conclude that a|-63, as -63 is divisible by a.

Answer: Therefore, if a|-7 and a|9, then a|-63.

Let's start with the first statement, a|-7. This means that a is a divisor of -7, or -7 is divisible by a. We can express this using the equation -7 = ak, where k is an integer.

Similarly, the second statement, a|9, means that a is a divisor of 9, or 9 is divisible by a. We can express this using the equation 9 = am, where m is an integer.

To prove that a|-63, we need to show that -63 is divisible by a. We can express this as -63 = al, where l is an integer.

Now, let's substitute the values of -7 and 9 from the previous equations into the equation for -63.

-7 = ak

9 = am

Multiplying the two equations together, we get:

(-7)(9) = (ak)(am)

-63 = (ak)(am)

-63 = a^2(km) (since we can multiply the exponents)

Since km is an integer (km = n, where n is an integer), we can rewrite the equation as:

-63 = a^2n

Therefore, we can conclude that a|-63, as -63 is divisible by a.

Answer: Therefore, if a|-7 and a|9, then a|-63.

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