Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 62

HARD

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IMPORTANT

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How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

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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 63

How many positive integer cubes divide

3! \cdot 5! \cdot 7! ?

MEDIUM

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 64

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is

6

. How many two-digit numbers have this property?

HARD

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IMPORTANT

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

For how many positive integers

n

does

1 + 2 + \dots + n

evenly divide from

6 n ?

HARD

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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 ?

HARD

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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})

?