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

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IMPORTANT

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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 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 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

?

HARD

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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

?

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 74

Given that

3^{8} \cdot 5^{2} = a^{b},

where both a and

b

are positive integers, find the smallest possible value for

(\frac{a + b + 93}{25})

.

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 75

What is the largest integer that is a divisor of

(n + 1) (n + 3) (n + 5) (n + 7) (n + 9)

for all positive even integers

n

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 76

The product of three consecutive positive integers is

8

times their sum. What is the sum of their squares?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 77

Let

P (n)

and

S (n)

denote the product and the sum, respectively, of the digits of the integer

n .

For example,

P (23) = 6

and

S (23) = 5 .

Suppose

N

is a two-digit number such that

N = P (N) + S (N) .

What is the units' digit of

N

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 78

The mean of three numbers is

10

more than the least of the numbers and

15

less than the greatest. The median of the three numbers is

5 .

What is their sum?