Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 7

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Some prime numbers are $1$ more than a square, as in $4^{2} + 1 = 17$ . How many primes $p \leq 100$ are of the form $p = n^{2} + 1$ ?

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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 8

How many prime numbers

x

satisfy

\sqrt{300} < x < \sqrt{700}

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 9

The number of pairs of prime numbers

(p, q)

satisfying the condition

\frac{51}{100} < \frac{1}{p} + \frac{1}{q} < \frac{5}{6}

will be

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 11

The number of positive integers

{}^{'}n^{'}

for which

3 n - 4, 4 n - 5

and

5 n - 3

are all primes is

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

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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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})

.

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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 ?

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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 ?