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How many ordered pairs $(a, b)$ of positive integers with $a < b$ and $100 \leq a, b \leq 1000$ satisfy $gcd (a, b) : lcm (a, b) = 1 : 495 ?$

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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 (11-08-2019)>Q 11

Consider the set $E$ of all natural numbers $n$ such that when divided by $11, 12, 13,$ respectively, the remainders, in that order, are distinct prime numbers in an arithmetic progression. If $N$ is the minimum number in $E$ find the sum of digits of $N$ .

Note: In PRMO-2019, original question was asking about maximum number in $E$ , which couldn't be calculated. So, this was the bonus question. Only minimum value can be calculated.

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2019 (11-08-2019)>Q 12

Consider the set

E = {5, 6, 7, 8, 9} .

For any partition

{A, B}

E,

with both

A

and

B

non-empty, consider the number obtained by adding the product of elements of

A

to the product of elements of

B .

Let

N

be the largest prime number among these numbers. Find the sum of the digits 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 (11-08-2019)>Q 13

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 .

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 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}

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 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

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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 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?

How many ordered pairs a,b of positive integers with a<b and 100≤a, b≤1000 satisfy gcda,b:lcma,b=1:495?

Important Questions on Number Theory

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How many ordered pairs $(a, b)$ of positive integers with $a < b$ and $100 \leq a, b \leq 1000$ satisfy $gcd (a, b) : lcm (a, b) = 1 : 495 ?$