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Chapter 6: Problem 457

For all \(n \in N-\\{1\\}, 7^{2 n}-48 n-1\) is divisible by (a) 25 (b) 26 (c) 1234 (d) 2304

Short Answer

Expert verified
The given expression \(7^{2n} - 48n - 1\) is divisible by 26 for all \(n \in N - \{1\}\). Therefore, the correct answer is (b) 26.

Step by step solution

01

Test Divisibility by 25

Let's analyze if \(7^{2n} - 48n - 1\) is divisible by 25. We can rewrite the expression in the following form: \(7^{2n} - 1\) - (\(48n\)). Notice that \(7^{2n} - 1\) is \((7^{n})^{2} - 1^{2}\), which is a difference of squares. We can factor this expression as \((7^{n} + 1)(7^{n} - 1)\). Let's analyze its divisibility by 25. We can now try some values for n and see whether the expression is divisible by 25: For n = 2: \(7^{2(2)} - 48(2) - 1 = (7^{4} - 1) - 96 = (2400) - 96 = 2304\) For n = 3: \(7^{2(3)} - 48(3) - 1 = (7^{6} - 1) - 144 = 117649 - 144 = 117505\) In both cases, the expression is not divisible by 25. Therefore, option (a) is not correct.
02

Test Divisibility by 26

Next, let's analyze the divisibility of \(7^{2n} - 48n - 1\) by 26. Let's apply the same test as before, by testing some values for n: For n = 2: \(7^{2(2)} - 48(2) - 1 = 2304\), which is divisible by 26. For n = 3: \(7^{2(3)} - 48(3) - 1 = 117505\), which is also divisible by 26. We can see that for both n = 2 and n = 3, the expression is divisible by 26. Thus, option (b) is correct. However, let's move on to options (c) and (d) to ensure that the others are not divisible.
03

Test Divisibility by 1234

Now we will analyze the divisibility of \(7^{2n} - 48n - 1\) by 1234. For n = 2, our expression is equal to 2304, which is not divisible by 1234. Therefore, option (c) is not correct.
04

Test Divisibility by 2304

Finally, let's analyze the divisibility of \(7^{2n} - 48n - 1\) by 2304. For n = 2, our expression is equal to 2304, which is divisible by 2304. However, we should test another value for n to ensure the divisibility doesn't hold for all values of n: For n = 3: \(7^{2(3)} - 48(3) - 1 = 117505\), which is not divisible by 2304. Thus, option (d) is not correct. Based on our analysis, we have found that the given expression is divisible by 26 for all \(n \in N - \{1\}\). Therefore, the correct answer is (b) 26.

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