Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 12

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$m, n$ are natural numbers. The number of pairs $(m, n)$ for which $m^{2} + n^{2} + 2 m n - 2013 m - 2013 n - 2014 = 0$ is

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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 13

Suppose that

a, b, c

are distinct numbers such that

{(b - a)}^{2} - 4 (b - c) (c - a) = 0

, find the value of

(\frac{b - c}{c - a})

.

EASY

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 14

Suppose that

p

and

q

are prime numbers

(p > q)

and they are roots of the equation

x^{2} - 99 x + m = 0

for some

m .

What is the value of

p - q ?

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 15

What is the largest possible prime value of

n^{2} - 12 n + 27,

where

n

ranges over all positive integers?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 17

Let

A

be the set of positive integers that have no prime factors other than

2, 3

or

5

. The infinite sum

\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \dots . .

of the reciprocals of the elements of

A

can be expressed as

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. What is

m + n ?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 18

For some positive integer

n

, the number

110 n^{3}

has

110

positive integer divisors, including

1

and the number

110 n^{3}

. The number

81 n^{4}

have

D

positive integer divisors. What is

(\frac{D}{5}) ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 19

What is the minimum number of digits to the right of the decimal point needed to express the fraction

\frac{123456789}{2^{26} \cdot 5^{4}}

as a decimal?

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

How many positive two-digit integers are factors of

2^{24} - 1

?

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IMPORTANT

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