Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 38

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There are $p$ pairs of positive integers $(a, b)$ such that $gcd (a, b) = 1$ and $\frac{a}{b} + \frac{14 b}{9 a}$ is an integer. What is $7 \times p ?$

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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 - 2.3>Q 39

Let

a, b, c, d, e, f, g

and

h

be distinct elements in the set

\{- 7, - 5, - 3, - 2, 2, 4, 6, 13\} .

What is the minimum possible value of

{(a + b + c + d)}^{2} + {(e + f + g + h)}^{2}

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 40

There are

88

numbers

a_{1}, a_{2}, a_{3}, \dots, a_{88}

and each of them is either equals to

- 3

or

- 1 .

Given that

a_{1}^{2} + a_{2}^{2} + \dots + a_{88}^{2} = 280

and

a_{1}^{4} + a_{2}^{4} + \dots + a_{88}^{4} = 10^{3} a + b .

Find the product of

a

and

b

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 41

The difference between the highest common factor and the lowest common multiple of

x

and

18

is

120 .

Find the value of

x .

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 42

Given that

n

is a ten-digit number in the form

\bar{2007 x \times 2008 y}

where

x

and

y

can be any of the digits

0, 1, 2, \dots,

9 .

How many such numbers

n

are there that are divisible by

33

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 43

How many positive integers

n,

where

10 \leq n \leq 100,

are there such that

(\frac{n^{2} - 9}{n^{2} - 7})

is a fraction in its lowest terms?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 44

Let

n

be a positive integer such that

n^{2} + 19 n + 48

is a perfect square. Find the value of

n

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 45

Find the smallest integer

n

such that

n (\sqrt{101} - 10) > 1

.

EASY

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 46

Find the smallest positive integer

N

such that

2^{n} > n^{2}

for every integer

n

in

\{N, N + 1, N + 2, N + 3, N + 4\}