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Chapter 7: Problem 517

If \(a, b \in N, a \neq b\) then for \(r \in N, a^{n}-b^{n}\) is divisible by (a) \(a-b\) (b) \(\mathrm{b}-\mathrm{a}\) (c) both (a) and (b) (d) None of these

Short Answer

Expert verified
Using the factorization of the difference of two powers, it can be shown that \(a^n - b^n\) is divisible by both \(a - b\) and \(b - a\). Therefore, the correct answer is (c) both (a) and (b).

Step by step solution

01

Write the factorization of the difference of two powers

The difference of two powers can be factored as follows: \[a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \cdots + b^{n-1})\]
02

Analyze the divisibility by \(a - b\)

As shown in Step 1, \(a - b\) is already a factor of \(a^n - b^n\), since the entire expression can be factored into two terms, one of which is \((a - b)\). Therefore, \(a^n - b^n\) is divisible by \(a - b\).
03

Analyze the divisibility by \(b - a\)

Notice that we can rewrite \(b - a\) as \(-(a - b)\). Thus, if we factor \(-1\) out of the expression in Step 1, we have: \[a^n - b^n = -(b - a)(-a^{n-1} - a^{n-2}b - a^{n-3}b^2 - \cdots - b^{n-1})\] Now, it is clear that \(b - a\) is also a factor of \(a^n - b^n\). Therefore, \(a^n - b^n\) is divisible by \(b - a\) as well.
04

Conclusion

Since \(a^n - b^n\) is divisible by both \(a - b\) and \(b - a\), the correct answer is (c) both (a) and (b).

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Most popular questions from this chapter

\(17^{\text {th }}\) term and \(18^{\text {th }}\) term are equal in expansion of \((2+x)^{40}\) then \(\mathrm{x}=\) (a) \((17 / 24)\) (b) \((17 / 12)\) (c) \((34 / 13)\) (d) \((34 / 15)\)

\(\mathrm{s}(\mathrm{k}): 1+3+5+\ldots+(2 \mathrm{k}-1)=3+\mathrm{k}^{2}\) then which statement is true ? (a) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (b) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (c) s (1) is true (d) Result is proved by Principle of Mathematical induction

th term is constant term in expansion of \(\left[\left(3 / x^{2}\right)+(\sqrt{x} / 3)\right]^{10}, x \neq 0\) (a) 4 (b) 7 (c) 8 (d) 9

Let \(\mathrm{x}>-1\) then statement \((1+\mathrm{x})^{\mathrm{n}}>1+\mathrm{n} \mathrm{x}\) is true for (a) \(\forall \mathrm{n} \in \mathrm{N}\) (b) \(\forall n>1\) (c) \(\forall \mathrm{n}>1\) and \(\mathrm{x} \neq 0\) (d) \(\forall \mathrm{n} \in \mathrm{R}\)

Constant term in expansion of \([1+(2 / \mathrm{x})-(2 / \mathrm{x})]^{4}\) is ........ (a) 5 (b) \(-5\) (c) 4 (d) \(-4\)
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