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Prove the following theorem: for every integer n, there is positive integer a and negative integer b for which a +b = n. (The easiest way to do this is by cases_ ...

Question

Prove the following theorem: for every integer n, there is positive integer a and negative integer b for which a +b = n. (The easiest way to do this is by cases_ There are three: n is positive, n = 0, 0r n is negative: In all cases; your a and b will depend on n!)

Prove the following theorem: for every integer n, there is positive integer a and negative integer b for which a +b = n. (The easiest way to do this is by cases_ There are three: n is positive, n = 0, 0r n is negative: In all cases; your a and b will depend on n!)



Answers

For any integers a and $b, b$ is a factor of a if there exists an integer $c$ such that $a=b c .$ Prove each of the following for any natural number $n$.
3 is a factor of $n^{3}+2 n$

When an equals one, the statement says that three is a factor of three which is obviously true. Next we assume this statement is true for and ecos K. Then for ecos K plus one, it becomes okay just one two. My nurse, Cape cod swat press three and this equals K cubed plus three K squared. Because three cake plus one. My nurse K -1 Plus three. This one in this negative one canceled, enhance the vehicles. Okay cubed minus K plus three then plus three times K squared plus K. Note that since the statement is true for any calls cake, The father term is a factor of three and The second term is also affect all three. And hence This expression is a factual three, which means the statement is true for Ecos K platform. Hence by mathematical induction, the statement is true for any end, which is a polarity of integer.

When an equals one, the statement says that three is a factor of three which is obviously true. Next we assume this statement is true for and ecos K. Then for ecos K plus one, it becomes okay just one two. My nurse, Cape cod swat press three and this equals K cubed plus three K squared. Because three cake plus one. My nurse K -1 Plus three. This one in this negative one canceled, enhance the vehicles. Okay cubed minus K plus three then plus three times K squared plus K. Note that since the statement is true for any calls cake, The father term is a factor of three and The second term is also affect all three. And hence This expression is a factual three, which means the statement is true for Ecos K platform. Hence by mathematical induction, the statement is true for any end, which is a polarity of integer.

First But only Coast one. The statement says that A plus B is a factor of a plastic which is obviously true. Next we assume that the statement is true for ecos cake for N equals K plus one. It becomes A to a power of two K plus one plus B. To a power of two K plus one yukos A squared times A. To the power of okay minus one plus pay squared times B to the power of UK miners, miners, B squared times A choose power of two K minus one, then plus B squared Times A. to the power of to K -1. And then it becomes a squared miners base square Times Age or Power of two K Miners. Because B squared A. To the power of joking miners one plus B to the power of to k minus one. Yeah. And it becomes A plus B times a minus B Times Age. Power to carry -1 Plus B Squared Times Age part of UK -1 Plus B. True power of two K miners one, the Stearman says the heart, it is true for an eco's cake, enhance. A plus B is a factor of the second term and a plus B is also a factor of the first term enhance A plus B in the factor of this expression, which means the statement is true for vehicles K plus one, the US. According to mathematical induction, the statement is true every end that is a positive integer.

First But only Coast one. The statement says that A plus B is a factor of a plastic which is obviously true. Next we assume that the statement is true for ecos cake for N equals K plus one. It becomes A to a power of two K plus one plus B. To a power of two K plus one yukos A squared times A. To the power of okay minus one plus pay squared times B to the power of UK miners, miners, B squared times A choose power of two K minus one, then plus B squared Times A. to the power of to K -1. And then it becomes a squared miners base square Times Age or Power of two K Miners. Because B squared A. To the power of joking miners one plus B to the power of to k minus one. Yeah. And it becomes A plus B times a minus B Times Age. Power to carry -1 Plus B Squared Times Age part of UK -1 Plus B. True power of two K miners one, the Stearman says the heart, it is true for an eco's cake, enhance. A plus B is a factor of the second term and a plus B is also a factor of the first term enhance A plus B in the factor of this expression, which means the statement is true for vehicles K plus one, the US. According to mathematical induction, the statement is true every end that is a positive integer.


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