Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 65

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IMPORTANT

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For how many positive integers $n$ does $1 + 2 + \dots + n$ evenly divide from $6 n ?$

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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 - 2.3>Q 66

For each positive integer

m > 1,

let

P (m)

denote the greatest prime factor of

m .

For how many positive integers

n

is it true that both

P (n) = \sqrt{n}

and

P (n + 48) = \sqrt{n + 48} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 67

Suppose that

4^{a} = 5, 5^{b} = 6, 6^{c} = 7

and

7^{d} = 8

. What is

4 \times a \times b \times c \times d ?

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 68

All of David's telephone numbers have the form

555 - a b c - d e f g

, where

a, b, c, d, e, f

and

g

are distinct digits and in increasing order and none is either

0

or

1 .

How many different telephone numbers can David have?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 69

For how many positive integers

n

less than or equal to

24

is

n!

evenly divisible by

1 + 2 + \dots + n ?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 70

Let

x

and

y

be two-digit integers such that

y

is obtained by reversing the digits of

x .

Suppose that the integers

x

and

y

satisfy

x^{2} - y^{2} = m^{2}

for some positive integer

' m' .

What is the value of

(\frac{x + y + m}{11})

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 71

A subset '

B

' of the set of integers from

1

to

100

inclusive, has the property that no two elements of

B

sum to

125 .

What is the maximum possible number of elements in

B

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 72

Let

n

be a

5

-digit number, and let

q

and

r

be the quotient and remainder, respectively, when

n

is divided by

100 .

Let the total number of value of

n

for which

q + r

divisible by

11

is

P .

Then find the last two digits of

P .

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 73

Let

F (x)

denote the sum of the digits of the positive integer

x .

For example,

F (8) = 8

and

F (123) = 1 + 2 + 3 = 6 .

For how many two-digit values of

x

is

F (F (x)) = 3

?