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Given that $x y z = 2014,$ and $x, y$ and $z$ are positive integers such that $x < y < z,$ how many possible triples $(x, y, z)$ are there?

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Important Questions on Number System

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 92

In the following equation, each of the letters represents uniquely a different digit in base ten:

(Y E) \cdot (M E) = T T T

The sum

E + M + T + Y

equals

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 93

Let

a, b, c,

and

d

be positive integers such that

gcd (a, b) = 24, gcd (b, c) = 36, gcd (c, d) = 54,

and

70 < gcd (d, a) < 100 .

Then, find the prime number which must be a divisor of

a

apart from

2

and

3

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 94

For a positive integer

n

and non-zero digits

a, b,

and

c,

let

A_{n}

be the

n

-digit integer each of whose digits is equal to

a;

let

B_{n}

be the

n

-digit integer each of whose digits are equal to

b,

and let

C_{n}

be the

2 n

-digit (not

n

-digit) integer each of whose digits are equal to

c .

What is the greatest possible value of

a + b + c

for which there are at least two values of

n

such that

C_{n} - B_{n} = A_{n}^{2} ?

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 95

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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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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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

Given that xyz=2014, and x, y and z are positive integers such that x<y<z, how many possible triples x,y,z are there?

Important Questions on Number System

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Given that $x y z = 2014,$ and $x, y$ and $z$ are positive integers such that $x < y < z,$ how many possible triples $(x, y, z)$ are there?