Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 11

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Let $p$ be a prime number such that the next larger number is a perfect square. Find the sum of all such prime numbers. (For example, if you think that $11$ and $13$ are two such prime numbers, then the sum is $24$ .)

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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 - 2.6>Q 12

Find $\frac{N - 2}{100},$ where $N$ is the greatest integer such that $N \leq \sqrt{2007^{2} - 20070 + 31}$ .

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 13

Find the sum of all the digit of largest integer

N

such that both

N + 496

and

N + 224

are perfect squares.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 14

The sum of

18

consecutive positive integers is a perfect square. For the smallest possible value of this sum, find the sum of first two integers ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 15

Let

n

be the smallest positive integer such that

n

is divisible by

20, n^{2}

is a perfect cube and

n^{3}

is a perfect square. The number

n

contains

N

digits. What is

N^{2}

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 16

How many positive integers

n

are there such that

7 n + 1

is a perfect square and

3 n + 1 < 2008

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 17

The product

1 \times 2 \times 3 \times . . . . . . . . . \times n

is denoted by

n!

. For example

4! = 1 \times 2 \times 3 \times 4 = 24

. Let

M = 1! \times 2! \times 3! \times 4! \times 5! \times 6! \times 7! \times 8! \times 9!

. Now the number of factors of

M

those are perfect squares is

21 b

, then find the value of

b

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 18

Find the number of ordered pairs

(x, y),

where

x

is an integer and

y

is a perfect square such that

y = {(x - 90)}^{2} - 4907

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.6>Q 19

The number

25^{64} \cdot 64^{25}

is the square of a positive integer

N

. In decimal representation, the sum of the digits of

N

is