Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 19

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IMPORTANT

Earn 100

Find the last two digits (in order) of $7^{3^{2005}} .$

100% studentsanswered this correctly

Important Questions on Number System

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 20

Find the smallest two-digit number

N

such that the sum of digits of

10^{N} - N

is divisible by

170 .

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 21

Let

N = 12345678910111213 \dots \dots 4344

be the

79

digit number obtained that is formed by writing the integers from

1

to

44

in order, one after the other. The remainder when

N

is divided by

45

is

r .

What is

r^{2} ?

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 22

Let

k = 2008^{2} + 2^{2008} .

What is the square of the units digit of

k^{2} + 2^{k} ?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 23

Anna randomly picked five integers from the following list

53, 62, 66, 68, 71, 82, 89

and discover that the average value of the five integers she picked is still an integer. If two of the integers she picked were

62

and

89,

find the mean of the remaining three integers.

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 24

If the remainder when

(1! \times 1) + (2! \times 2) + (3! \times 3) + \dots + (286! \times 286)

is divided by

2^{4} \times 5^{3}

is

r

, then

r - 1900

is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 25

What is the remainder when

3^{0} + 3^{1} + 3^{2} + \dots + 3^{2009}

is divided by

8 ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 26

How many distinct four-digit numbers are divisible by

3

and have

23

as their last two digits?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.5>Q 27

What is the last two digits of

2015^{2016} - 2017 ?