If n is a natural number, then 8n – 3n is always divisible by ……

1.	If n is a natural number, then 8n – 3n is always divisible by …………….. A) 3 B) 5 C) 8 D) 11
Answer» Correct option is (B) 5 For n = 1, \(8^n-3^n=8^1-3^1\) = 8 - 3 = 5 which is divisible by 5 (not divisible by 3, 8 or 11) Alternative :- \(8^n-3^n=(3+5)^n-3^n\) \(=3^n+\,^nC_1\,3^{n-1}\times5+\,^nC_2\,3^{n-2}\times5^2+....+5^n-3^n\) (By using binomial expansion) \(=5\,(^nC_1\times3^{n-1}+\,^nC_2\times3^{n-2}\times5+....+5^{n-1})\) = 5q, where \(q=\,^nC_1\times3^{n-1}+\,^nC_2\times3^{n-2}\times5+....+5^{n-1}\) \(\in I\,or\,Z.\) \(\therefore\) \(8^n-3^n\) is a multiple of 5. Thus, \(8^n-3^n\) is divisible by 5. Correct option is B) 5

If n is a natural number, then 8n – 3n is always divisible by …………….. A) 3 B) 5 C) 8 D) 11

Answer»

Correct option is (B) 5

For n = 1, \(8^n-3^n=8^1-3^1\)

= 8 - 3 = 5 which is divisible by 5 (not divisible by 3, 8 or 11)

Alternative :-

\(8^n-3^n=(3+5)^n-3^n\)

\(=3^n+\,^nC_1\,3^{n-1}\times5+\,^nC_2\,3^{n-2}\times5^2+....+5^n-3^n\) (By using binomial expansion)

\(=5\,(^nC_1\times3^{n-1}+\,^nC_2\times3^{n-2}\times5+....+5^{n-1})\)

= 5q, where \(q=\,^nC_1\times3^{n-1}+\,^nC_2\times3^{n-2}\times5+....+5^{n-1}\) \(\in I\,or\,Z.\)

\(\therefore\) \(8^n-3^n\) is a multiple of 5.

Thus, \(8^n-3^n\) is divisible by 5.

Correct option is B) 5