Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 21

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IMPORTANT

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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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Important Questions on Number System

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IMPORTANT

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 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 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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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 34

Three positive integers, each greater than

1,

have a product of

27000

and are pairwise relatively prime. What is their mean to the nearest whole number?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 36

Let

a

and

b

be relatively prime integers with

a > b > 0

and

\frac{a^{3} - b^{3}}{{(a - b)}^{3}} = \frac{73}{3}

. What is

a - b

?