Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 25

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Determine the sum of digits of the largest prime number less than $5000$ of the form $a^{n} - 1,$ where $a$ and $n$ are positive integers and $n$ is greater than . $1$

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Important Questions on Number System

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 26

Let

f

be a real-valued function with the rule

f (x) = x^{3} + 3 x^{2} + 6 x + 14

defined for all real value of

x .

It is given that

a

and

b

are two real numbers such that

f (a) = 1

and

f (b) = 19

. Find the value of

{(a + b)}^{2}

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 27

Positive integers

a, b

and

2009

with

a < b < 2009

form a geometric sequence with an integer ratio. What is

a

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 28

For

k > 0,

let

I_{k} = 10 \dots 064,

where there are

k

zeros between

1

and

6

. Let

N (k)

be the number of factors of

2

in the prime factorization of

I_{k}

. What is the maximum value of

N (k)

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 29

Suppose that

m

and

n

are positive integers such that

75 m = n^{3}

. What is the minimum possible value of

m + n

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 30

Let

n

be the largest integer that is the product of exactly

3

distinct prime numbers

d, e

and

10 d + e

, where

d

and

e

are single digits. What is the sum of the digits of

n

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 31

The digits

1, 2, 3, 4, 5, 6, 7

and

9

are used to form four two-digit prime numbers, with each digit used exactly once, then the sum of these four prime numbers is given by

10 A

. Find the value of

A

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 32

The positive integers

A, B, A - B

and

A + B

are all prime numbers. The sum of these four prime numbers is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 33

Find the number of counterexamples to the statement:

"If $N$ is an odd positive integer, the sum of whose digits is $4$ and none of whose digits is $0,$ then $N$ is prime."