Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 12

HARD

IOQM - PRMO and RMO

IMPORTANT

Earn 100

Find the unit's digit of $(2013)^{2013} .$

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Important Questions on Number System

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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 13

Suppose that

n

is the product of three consecutive integers and that

n

is divisible by

7 .

Which of the following is not necessarily a divisor of

n ?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 14

Find the last two digits of

\frac{N}{1000},

if

N = 9^{100} - 1 .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 15

Find the sum of digits of the largest positive integer

n

such that

n!

ends with exactly

100

zeros.

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 16

Find the number of positive integers

n

such that

n + 2 n^{2} + 3 n^{3} + \dots + 2005 n^{2005}

is divisible by

n - 1 .

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.2>Q 17

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 $11$ .

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 ?