MEDIUM

IOQM - PRMO and RMO

IMPORTANT

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Let $n$ be the number $\underset{2006 9' s}{\underset{⏟}{(999999999 . . . 999)^{2}}} - \underset{2006 6^{'} s}{\underset{⏟}{(666666666 \dots 666)^{2}}} .$ Find the remainder when $n$ is divided by $11$ .

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Important Questions on Number System

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 18

What is the sum of all the digits of the largest positive integer

n

for which

n^{3} + 2006

is divisible by

n + 26 ?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 19

Find the smallest positive integer

n

such that

n (n + 1) (n + 2)

is divisible by

247 .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 21

Let

n

denotes the smallest positive integer that is divisible by both

4

and

9,

and whose base-

10

representation consists of only

4' s

and

9' s,

with at-least one of each. What will be the sum of the last four digits of

n ?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 22

n

-digit positive integer is cute if its

n

digits are an arrangement of the set

{1, 2, 3, \dots, n}

and its first

k

digits form an integer that is divisible by

k,

for

k = 1, 2, 3, \dots, n .

For example,

321

is a cute

3

-digit integer because

1

divides

3, 2

divides

32

and

3

divides

321 .

How many cutes

6

-digit integers are there?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 23

Consider the non-decreasing sequence of positive integers

$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, \dots \dots .$ in which the $n^{th}$ positive integer appears $n$ times. The remainder when the $2019^{th}$ term is divided by $5$ is

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 24

Mary chose an even

4

-digit number

n .

She wrote down all the divisors of

n

in increasing order from left to right:

1, 2, \dots, \frac{n}{2}, n .

At some moment, Mary wrote

323

as a divisor of

n .

If the smallest possible value of the next divisor written to the right of

323

N .

Find the value of

\frac{N}{10} .

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 25

The largest divisor of

2014000000

is itself. If its fifth largest divisor is

X .

Find sum of all the digits of

X .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 26

How many three-digit numbers are not divisible by

5,

have digits that sum to less than

20,

and have the first digit equal to the third digit?

Let n be the number (999999999...999)2⏟2006 9's-(666666666…666)2⏟2006 6's. Find the remainder when n is divided by 11.

Important Questions on Number System

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Let $n$ be the number $\underset{2006 9' s}{\underset{⏟}{(999999999 . . . 999)^{2}}} - \underset{2006 6^{'} s}{\underset{⏟}{(666666666 \dots 666)^{2}}} .$ Find the remainder when $n$ is divided by $11$ .