Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 51

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Let $S$ be the set of all integers $n$ such that $\frac{8 n^{3} - 96 n^{2} + 360 n - 400}{2 n - 7}$ is an integer. Find the value of $\sum_{n \in S} |n|$ .

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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 52

Find the sum of all positive integers

p

such that the expression

(x - p) (x - 13) + 4

can be expressed in the form

(x + q) (x + r)

for distinct integers

q

and

r .

HARD

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 53

Let

P_{k} = 1 + \frac{1}{k} - \frac{1}{k^{2}} - \frac{1}{k^{3}},

where

k

is a positive integer. Find the sum of all the digits of least positive integer

n

such that the product

P_{2} P_{3} \dots P_{n}

exceeds

2010 .

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 54

What is the sum of the digits of the square of

111, 111, 111 ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 55

Let

a, b, c,

and

d

be real numbers with

|a - b| = 2, |b - c| = 3

and

|c - d| = 4 .

What is the sum of all possible values of

|a - d|

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 56

Let

k = 2008^{2} + 2^{2008} .

What is the units digit of

k^{2} + 2^{k} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 57

The numbers from

1

to

8

are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 58

The Saxena family consists of a mother, a father, and some children. The average age of the members of the family is

20

, the father is

48

years old, and the average age of the mother and children is

16

. How many children are in the family?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 59

How many ordered pairs

(m, n)

of positive integers, with

m > n,

have the property that their squares differ by

96 ?