### Mathematical induction and Divisibility problems:

**Ques. **For all positive integral values of *n*, 3^{2n} – 2n + 1 is divisible by

(a) 2

(b) 4

(c) 8

(d) 12

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**Ques. **If n ∈ N, then x^{2n – 1} + y^{2n – 1} is divisible by

(a) x + y

(b) x – y

(c) x^{2} + y^{2}

(d) x^{2} + xy

**Ques. **If n ∈ N, then 7^{2n} + 2^{3n – 3}. 3^{n – 1} is always divisible by

(a) 25

(b) 35

(c) 45

(d) None of these

**Ques. **If *p* is a prime number, then n^{p} – n is divisible by *p* when *n* is a

(a) Natural number greater than 1

(b) Irrational number

(c) Complex number

(d) Odd number

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**Ques. **For every natural number *n*, *n*(*n* + 1) is always

(a) Even

(b) Odd

(c) Multiple of 3

(d) Multiple of 4

**Ques. **If n ∈ N, then 11^{n + 2} + 12^{2n + 1} is divisible by

(a) 113

(b) 123

(c) 133

(d) None of these

**Ques. **For every natural number *n*, n(n^{2} – 1) is divisible by

(a) 4

(b) 6

(c) 10

(d) None of these

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**Ques. **The statement *P*(*n*) “1 x 1! + 2 x 2! + 3 x 3! + … + n x n! = (n + 1)! – 1” is

(a) True for all *n* > 1

(b) Not true for any *n
*(c) True for all

*n*∈

*N*

(d) None of these

**Ques. **For every natural number *n
*(a) n > 2

^{n}

(b) n < 2

^{n }(c) n = 2

(d) n = 2n

^{2}

**Ques. **The remainder when 5^{99} is divided by 13 is

(a) 6

(b) 8

(c) 9

(d) 10

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**Ques. **For each n ∈ N, the correct statement is

(a) 2^{n} < n

(b) n^{2} > 2n

(c) n^{4} < n^{n}

(d) 2^{3n} > 7n + 1

**Ques. **10^{n} + 3(4^{n+2}) + 5 is divisible by (n ∈ N)

(a) 7

(b) 5

(c) 9

(d) 17

**Ques. **For natural number *n*, (n!)^{2} > n^{n}, if

(a) *n* > 3

(b) *n* > 4

(c) *n* ³ 4

(d) *n* ³ 3

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**Ques. **For natural number *n*, 2^{n} (n – 1)! < n^{n}, if

(a) *n* < 2

(b) *n* > 2

(c) *n* ³ 2

(d) Never

**Ques. **For positive integer *n*, 10^{n – 2} > 81n, if

(a) *n* > 5

(b) *n* ³ 5

(c) *n* < 5

(d) *n* > 6

**Ques. **When 2^{301} is divided by 5, the least positive remainder is

(a) 4

(b) 8

(c) 2

(d) 6

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**Ques. **For every positive integer *n*, 2^{n} < n! when

(a) *n* < 4

(b) *n* ³ 4

(c) *n* < 3

(d) None of these

**Ques. **x(x^{n–1} – na^{n–1}) + a^{n}(n–1) is divisible by (x – a)^{2} for

(a) *n* > 1

(b) *n* > 2

(c) All *n* ∈ *N*

(d) None of these

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**Ques. **For every positive integral value of *n*, 3^{n} > n^{3} when

(a) *n* > 2

(b) *n* ³ 3

(c) *n* ³ 4

(d) *n* < 4