Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 35

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There are total $P$ positive integers value of $n,$ so that $\frac{n}{30 - n}$ is also a positive integer. What is $P^{2} ?$

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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 36

Let

S

be the set of positive integers

n

for which

\frac{1}{n}

has the repeating decimal representation

0 . \bar{a b} = 0 . a b a b a b a b . . . . .

with

a

and

b

different digits. The sum of the elements of

S

is a three-digit number

a b c

. What is the value of

\frac{a b c}{11} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 37

Let

a, b,

and

c

be digit with

a \neq 0 .

The three-digit integer

a b c

lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer

a c b

lies two thirds of the way between the same two squares. What is

a + b + c ?

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

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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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?