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IMPORTANT

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Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set ${1, 2, 3, \dots \dots, n}$ can be partitioned into $3$ subsets with equal sums. Find the number of elements of $E .$

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Important Questions on Number Theory

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IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 14

Consider the sequence of numbers

[n + \sqrt{2 n} + \frac{1}{2}]

for

n \geq 1,

where

[x]

denotes the greatest integer not exceeding

x

. If the missing integers in the sequence are

n_{1} < n_{2} < n_{3} < \dots . .,

then find

n_{12}

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IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 15

Let

a_{1} = 24

and form the sequence

a_{n}, n \geq 2

a_{n} = 100 a_{n - 1} + 134

. The first few terms are

24, 2534, 253534, 25353534, \dots . . . .

. What is the least value of

n

for which

a_{n}

is divisible by

99

MEDIUM

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IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 16

Let

N

be the smallest positive integer such that

N + 2 N + 3 N + . . . . . . . . . + 9 N

is a number all whose digits are equal. What is the sum of the digits of

N

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 17

Let

s (n)

denote the sum of the digits of a positive integer

n

in base

10

. If

s (m) = 20

and

s (33 m) = 120,

what is the value of

s (3 m)

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 18

Let

F_{k} (a, b) = {(a + b)}^{k} - a^{k} - b^{k}

and let

S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

. For how many ordered pairs

(a, b)

with

a, b \in S

and

a \leq b

\frac{F_{5} (a, b)}{F_{3} (a, b)}

an integer?

HARD

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IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 19

Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $\{10 a + b : 1 \leq a \leq 5, 1 \leq b \leq 5\}$ , where $a, b$ are integers, such that no two of the selected numbers have the same unit's digit and no two have the same ten's digit. What is the remainder when $N$ is divided by $73$ ?

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IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 20

Let

N

denote the number of all natural numbers

n

such that

n

is divisible by a prime

p > \sqrt{n}

and

p < 20 .

What is the value of

N ?

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 21

Let

a, b, c

be distinct positive integers such that

b + c - a, c + a - b

and

a + b - c

are all perfect squares. What is the largest possible value of

a + b + c

smaller than

100 ?

Let E denote the set of all natural numbers n such that 3<n<100 and the set {1,2,3,……,n} can be partitioned into 3 subsets with equal sums. Find the number of elements of E.

Important Questions on Number Theory

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Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set ${1, 2, 3, \dots \dots, n}$ can be partitioned into $3$ subsets with equal sums. Find the number of elements of $E .$