Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 28

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For $k > 0,$ let $I_{k} = 10 \dots 064,$ where there are $k$ zeros between $1$ and $6$ . Let $N (k)$ be the number of factors of $2$ in the prime factorization of $I_{k}$ . What is the maximum value of $N (k)$ ?

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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 - 2.1>Q 29

Suppose that

m

and

n

are positive integers such that

75 m = n^{3}

. What is the minimum possible value of

m + n

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 30

Let

n

be the largest integer that is the product of exactly

3

distinct prime numbers

d, e

and

10 d + e

, where

d

and

e

are single digits. What is the sum of the digits of

n

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 31

The digits

1, 2, 3, 4, 5, 6, 7

and

9

are used to form four two-digit prime numbers, with each digit used exactly once, then the sum of these four prime numbers is given by

10 A

. Find the value of

A

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 32

The positive integers

A, B, A - B

and

A + B

are all prime numbers. The sum of these four prime numbers is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 33

Find the number of counterexamples to the statement:

"If $N$ is an odd positive integer, the sum of whose digits is $4$ and none of whose digits is $0,$ then $N$ is prime."

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 34

Three positive integers, each greater than

1,

have a product of

27000

and are pairwise relatively prime. What is their mean to the nearest whole number?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 35

A positive integer

n

is nice if there is a positive integer

m

with exactly four positive divisors (including

1

and

m

) such that the sum of the four divisors is equal to

n

. How many numbers in the set

\{2010, 2011, 2012, \dots . . ., 2019\}

are nice?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 36

Let

a

and

b

be relatively prime integers with

a > b > 0

and

\frac{a^{3} - b^{3}}{{(a - b)}^{3}} = \frac{73}{3}

. What is

a - b

?