EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2018>Q 29

HARD

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IMPORTANT

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Let $T$ be the smallest positive integer which, when divided by $11, 13, 15$ leaves remainders in the sets ${7, 8, 9}, {1, 2, 3}, {4, 5, 6}$ , respectively. What is the sum of the squares of the digits of $T ?$

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Important Questions on Number Theory

MEDIUM

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IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2020>Q 1

If

a, b, c

are real numbers and

(a + b - 5)^{2} + (b + 2 c + 3)^{2} + (c + 3 a - 10)^{2} = 0

, find the integer nearest to

a^{3} + b^{3} + c^{3}

.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2020>Q 2

Find the number of positive integers

n

such that the highest power of

7

dividing

n!

is

8 .

EASY

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2020>Q 3

Find the largest

2

-digit number

N

which is divisible by

4,

such that all integral power of

N

end with

N .

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2020>Q 4

The prime numbers

a, b

and

c

are such that

a + b^{2} = 4 c^{2} .

Determine the sum of all possible values of

a + b + c .

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2020>Q 5

A five-digit number

n = \bar{a b c d e}

is such that when divided respectively by

2, 3, 4, 5, 6

the remainders are

a, b, c, d, e .

What is the remainder when

n

is divided by

100 ?

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2020>Q 6

Let

a, b, c

be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum

a + b + c .

Find this sum.

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 11 - Number Theory>IOQM - PRMO and RMO 2020>Q 7

For a natural number

n

, let

n^{'}

denote the number obtained by deleting zero digits, if any. (For example, if

n = 260, n^{'} = 26;

if

n = 2020, n^{'} = 22

). Find the number of

3

-digit numbers

n

for which

n^{'}

is a divisor of

n,

different from

n .

MEDIUM

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 8

Let the rational number

\frac{p}{q}

, be closest to but not equal to

\frac{22}{7}

among all rational numbers with denominator

< 100 .

What is the value of

p - 3 q ?