Chapter 6: Problem 485

\(10^{\mathrm{n}}+3\left(4^{\mathrm{n}+2}\right)+5\) is divisible by \((\mathrm{n} \in \mathrm{N})\) (a) 7 (b) 5 (c) 9 (d) 17

Short Answer

Expert verified

The expression \(10^n + 3\left(4^{n+2}\right) + 5\) is divisible by 9 (option c).

Step by step solution

Test with option (a) 7#

Calculate the expression modulo 7: \[10^n + 3\left(4^{n+2}\right) + 5 \equiv 3^n + 3(4^{n})\pmod{7}\] We will test first for n = 0, n = 1, and n = 2 (we are looking for a pattern): - For n = 0: \(3^0 + 3(4^0) \equiv 1 + 3 \equiv 4 \pmod{7}\) - For n = 1: \(3^1 + 3(4^1) \equiv 3 + 12 \equiv 1 \pmod{7}\) - For n = 2: \(3^2 + 3(4^2) \equiv 9 + 48 \equiv 0 \pmod{7}\) We found a pattern: for n = 2, the expression is divisible by 7. However, further trials would help to confirm if the pattern persists.

Test with option (b) 5#

Calculate the expression modulo 5: \[10^n + 3\left(4^{n+2}\right) + 5 \equiv 0 + 2\left(4^n\right) + 0 \equiv 2\left(4^n\right) \pmod{5}\] We will test first for n = 0, n = 1 and n = 2: - For n = 0: \(2(4^0) \equiv 2 \pmod{5}\) - For n = 1: \(2(4^1) \equiv 8 \equiv 3 \pmod{5}\) - For n = 2: \(2(4^2) \equiv 32 \equiv 2 \pmod{5}\) In this case, the expression is not divisible by 5 for any of our tested values of n. Thus, we can eliminate option (b).

Test with option (c) 9#

Calculate the expression modulo 9: \[10^n + 3\left(4^{n+2}\right) + 5 \equiv 1^n + 3\left(4^{n}\right) + 5 \equiv 3\left(4^n\right) + 6 \pmod{9}\] We will test first for n = 0, n = 1 and n = 2: - For n = 0: \(3(4^0) + 6 \equiv 3 + 6 \equiv 0 \pmod{9}\) - For n = 1: \(3(4^1) + 6 \equiv 12 + 6 \equiv 0 \pmod{9}\) - For n = 2: \(3(4^2) + 6 \equiv 48 + 6 \equiv 0 \pmod{9}\) In this case, the expression is divisible by 9 for all tested values of n. So, the divisibility by 9 is established.

Test with option (d) 17#

Calculate the expression modulo 17: \[10^n + 3\left(4^{n+2}\right) + 5 \equiv (-7)^n + 3\left(4^n\right)\left(4^2\right) + 5 \pmod{17}\] Again, we will test first for n = 0, n = 1 and n = 2: - For n = 0: \((-7)^0 + 3(4^0)(4^2) + 5 \equiv 1 + 48 + 5 \equiv 10 \pmod{17}\) - For n = 1: \((-7)^1 + 3(4^1)(4^2) + 5 \equiv -7 + 192 + 5 \equiv 4 \pmod{17}\) - For n = 2: \((-7)^2 + 3(4^2)(4^2) + 5 \equiv 49 + 3072 + 5 \equiv 8 \pmod{17}\) In this case, the expression is not divisible by 17 for any of our tested values of n. Thus, we can eliminate option (d) too. Based on our tests, the correct answer is option (c) 9. The expression \(10^n + 3\left(4^{n+2}\right) + 5\) is divisible by 9 for the tested values of n.

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\(10^{\mathrm{n}}+3\left(4^{\mathrm{n}+2}\right)+5\) is divisible by \((\mathrm{n} \in \mathrm{N})\) (a) 7 (b) 5 (c) 9 (d) 17

Short Answer

Step by step solution

Test with option (a) 7#

Test with option (b) 5#

Test with option (c) 9#

Test with option (d) 17#

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