Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 28

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Earn 100

$R$ non-negative integers can be written in the form $a_{7} \cdot 3^{7} + a_{6} \cdot 3^{6} + a_{5} \cdot 3^{5} + a_{4} \cdot 3^{4} + a_{3} \cdot 3^{3} + a_{2} \cdot 3^{2} + a_{1} \cdot 3^{1} + a_{0} 3^{0}$

where $a_{i} \in \{- 1, 0, 1\}$ for $0 \leq i \leq 7 .$ What is the last two digits of $R ?$

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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 29

How many odd positives

3

-digit integers are divisible by

3

but do not contain the digit

3 ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 30

Let

p

and

q

be positive integers such that

\frac{5}{9} < \frac{p}{q} < \frac{4}{7}

and

q

is as small as possible. What is

{(q - p)}^{2} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 31

A function

f

is defined recursively by

f (1) = f (2) = 1

and

f (n) = f (n - 1) - f (n - 2) + n

for all integers

n \geq 3 .

What is the sum of the digits of

f (2018) ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 32

Sunita has

30

thin rods, one each of every integer length from

1 cm

through

30 cm .

She places the rods with lengths

3 cm, 7 cm,

and

15 cm

on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 33

A five-digit palindrome is a positive integer with respective digits

a b c b a

, where

a

is non-zero. Let

S

be the sum of all five-digit palindromes. What is the sum of the digits of

S ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 34

The products

(8) (888 \dots . 8)

, where the second factor has

k

digits is an integer whose digits have a sum of

1000

. What is the sum of digits of

k ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 35

There are total

P

positive integers value of

n,

so that

\frac{n}{30 - n}

is also a positive integer. What is

P^{2} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 36

Let

S

be the set of positive integers

n

for which

\frac{1}{n}

has the repeating decimal representation

0 . \bar{a b} = 0 . a b a b a b a b . . . . .

with

a

and

b

different digits. The sum of the elements of

S

is a three-digit number

a b c

. What is the value of

\frac{a b c}{11} ?