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IMPORTANT

Earn 100

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $a \cdot b + 63 = 20 \cdot lcm (a, b) + 12 \cdot gcd (a, b)$

where $gcd (a, b)$ denotes the greatest common divisor of $a$ and $b,$ and $lcm (a, b)$ denotes their least common multiple?

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Important Questions on Number System

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 96

There are

10

horses, named Horse

1,

Horse

2, \dots,

Horse

10 .

They get their names from how many minutes it takes them to run one lap around a circular racetrack: Horse

k

runs one lap in exactly

k

minutes. At time

0

, all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time

S > 0,

in minutes, at which all

10

horses will again simultaneously be at the starting point is

S = 2520

. Let

T > 0

be the least time, in minutes, such that at least

5

of the horses are again at the starting point. What is the sum of the digits of

T ?

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IOQM - PRMO and RMO

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 97

How many ordered triples

(x, y, z)

of positive integers satisfy

lcm (x, y) = 72, lcm (x, z) = 600

and

lcm (y, z) = 900 ?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 98

For the greatest value of

x,

2^{x}

is a factor of

10^{1002} - 4^{501} .

Find the value of

\frac{x}{15} .

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 99

In base

10,

the number

2013

ends in the digit

3

. In base

9,

on the other hand, the same number is written as

(2676) 9

and ends in the digit

6 .

For how many positive integers

b

does the base-

b

representation of

2013

end in the digit

3

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 100

Real numbers

x

and

y

satisfy the equation

x^{2} + y^{2} = 10 x - 6 y - 34 .

What is

x + y

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 101

The number

2013

has the property that its units digit is the sum of its other digits, that is

2 + 0 + 1 = 3

. How many integers less than

2013

but greater than

1000

share this property?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 102

The number $2013$ is expressed in the form $2013 = \frac{(a_{1}!) (a_{2}!) \dots (a_{m}!)}{(b_{1}!) (b_{2}!) \dots (b_{n}!)}$

where $a_{1} \geq a_{2} \geq \dots \geq a_{m}$ and $b_{1} \geq b_{2} \geq \dots b_{n}$ are positive integers and $a_{1} + b_{1}$ is as small as possible. What is
$|a_{1} - b_{1}| ?$

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 103

Two integers have a sum of

26 .

When two more integers are added to the first two integers the sum is

41 .

Finally, when two more integers are added to the sum of the previous four integers the sum is

57 .

What is the minimum number of even integers among the

6

integers?

How many ordered pairs a,b of positive integers satisfy the equation a·b+63=20·lcma,b+12·gcda,b where gcda,b denotes the greatest common divisor of a and b, and lcma,b denotes their least common multiple?

Important Questions on Number System

Learn and Prepare for any exam you want !

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $a \cdot b + 63 = 20 \cdot lcm (a, b) + 12 \cdot gcd (a, b)$

where $gcd (a, b)$ denotes the greatest common divisor of $a$ and $b,$ and $lcm (a, b)$ denotes their least common multiple?