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The sum of two natural number is $100$ . Which of the following can be the greatest product of the numbers?

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Important Questions on Number System

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 12

A computer is printing a list of the seventh powers of all natural numbers, that is the sequence

1^{7}, 2^{7}, 3^{7}, \dots \dots

The number of terms (or numbers) between

5^{21}

and

4^{28}

are

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 13

For a positive integer

k,

we write

(1 + x) (1 + 2 x) (1 + 3 x) \dots (1 + k x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{k} x^{k}

where

a_{0}, a_{1}, \dots, a_{k}

are the coefficients of the polynomial. Find the sum of all the digits of the smallest possible value of

k

a_{0} + a_{1} + a_{2} + \dots + a_{k - 1}

is divisible by

2005

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 14

The smallest three-digit number $n$ such that if the three digits are $a, b$ and $c,$ then $n = a + b + c + a b + b c + a c + a b c$

Evaluate $\frac{n + 1}{5}$

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 15

What is the largest positive integer

n

satisfying

n^{200} < 5^{300} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 16

N

pieces of candy are made and packed into boxes, with each box containing

45

pieces. If

N

is a non-zero perfect cube and

45

is one of its factors, what is the least possible number of boxes that can be packed?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 17

For any non-empty finite set

A

of real numbers, let

s (A)

be the sum of the elements in

A

. There are exactly

61 3

-element subsets

A

\{1, \dots, 23\}

with

s (A) = 36 .

The total number of

3

-element subsets of

\{1, \dots, 23\}

with

s (A) < 36

is given by

N .

Then evaluate

\frac{N + 45}{25}

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 18

Let the sum of the digits of the given product is written as $A 8 B 6 C .$ Find $A + B + C$ .

$\underset{2007 times}{\underset{⏟}{(111111111 \dots 111)}} \times 2007$

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 19

When

2007

bars of soap are packed into

N

boxes of equal size, where

N

is an integer strictly between

200

and

300,

there are extra

5

bars remaining. Find last two digit of

N .

The sum of two natural number is 100. Which of the following can be the greatest product of the numbers?

Important Questions on Number System

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The sum of two natural number is $100$ . Which of the following can be the greatest product of the numbers?