Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 20

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How many positive two-digit integers are factors of $2^{24} - 1$ ?

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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 21

Let

N

be the positive integer for which the sum of its two smallest factors is

4

and the sum of its two largest factors is

204

. The value of

(\frac{N}{3})

is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 22

Let

S = {1, 2, 3, \dots, 20}

be the set of all positive integers from

1

to

20

. Suppose that

N

is the smallest positive integer such that exactly eighteen numbers from

S

are factors of

N

, and the only two numbers from

S

that are not factors of

N

are consecutive integers. Find the sum of the digits of

N

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 23

m

and

n

are two positive integers of reverse order (for example,

123

and

321

) such that

m n = 1446921630

. Now the value of

m + n

is given as

a b 4 b a

. Find the product of

a

and

b

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 24

Find the number of positive divisors of

{(2008^{3} + (3 \times 2008 \times 2009) + 1)}^{2}

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

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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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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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)

?