There are two prime numbers p so that 5p can be expressed in the form of -n25- for some positive integer n- where - denotes the greatest integer function- The sum of these two prime numbers is

5p=[n^25] for some primes p Remainder when n^2 divided by 5 must be 0, 1 and 4. So, nbsp;25p = n^2 or n^2 1 or n^2 4 If 25p = n^2 1 = (n 1) (n + 1), ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Direct Method of Validation

There are two...

Question

There are two prime numbers $p$ so that $5 p$ can be expressed in the form of $[\frac{n^{2}}{5}]$ for some positive integer $n,$ where $[.]$ denotes the greatest integer function. The sum of these two prime numbers is

52

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

50

No worries! We‘ve got your back. Try BYJU‘S free classes today!

48

No worries! We‘ve got your back. Try BYJU‘S free classes today!

54

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $52$
$5 p = [\frac{n^{2}}{5}]$ for some primes $p$
Remainder when $n^{2}$ divided by $5$ must be $0, 1$ and $4.$
So, $25 p = n^{2}$ or $n^{2} - 1$ or $n^{2} - 4$

If $25 p = n^{2} - 1 = (n - 1) (n + 1),$
then $n - 1 = 25$ or $n + 1 = 25$
as $n - 1$ and $n + 1$ both cannot be divisible by $5.$

$25 p = n^{2} - 4 = (n - 2) (n + 2)$
So, $n - 2 = 25$ or $n + 2 = 25$

Therefore, $n = 23, 24, 26, 27$ are possible values.
Correspondingly, the values of $p$ are $21, 23, 27$ and $29.$
$23$ and $29$ are prime.
Required sum $= 23 + 29 = 52$

Suggest Corrections

Similar questions

Let

p

be a prime number &

n

be a positive integer, then exponent of prime

p

n!

is denoted by

E_{p} (n!)

& is given by

E_{p} (n!) = [\frac{n}{p}] + [\frac{n}{p^{2}}] + [\frac{n}{p^{3}}] + . . . . . + [\frac{n}{p^{x}}]

where

x

is the largest positive integer such that

p^{x} \leq n < p^{x + 1}

and

[\cdot]

denotes the greatest integer

Again every natural number

N

can be expressed as the product of its prime factors given by

N = P_{1}^{k_{2}} P_{2}^{k_{2}} . . . . P_{r}^{k_{r}}

where

P_{1}, P_{2}, P_{3}, . . . . . . . P_{r}

are prime numbers &

k_{1}

are whole numbers.

The greatest integer

n

for which

77!

is divisible by

3^{n}

Q. Let

f (x) = [n + p sin x], x \in (0, π), n \in Z

and

p

is prime number, where

[.]

denotes the greatest integer function. Then, the number of points where

f (x)

is not differentiable, are

Q. If

p

is a prime number greater than

2

, then the difference

[{(2 + \sqrt{5})}^{p}] - 2^{p + 1}

, where

[.]

denotes greatest integer. is divisible by

Q. Let

f (x) = [n + p sin x], x ϵ (0, π), n ϵ Z

and

P

is prime number, where

[.]

denotes the greatest integer function. Then number of points where

f (x)

in not differentiable is ?

Numbers that can be expressed in the form $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ are called ___ numbers.

Join BYJU'S Learning Program

There are two prime numbers p so that 5p can be expressed in the form of [n25] for some positive integer n, where [.] denotes the greatest integer function. The sum of these two prime numbers is

There are two prime numbers $p$ so that $5 p$ can be expressed in the form of $[\frac{n^{2}}{5}]$ for some positive integer $n,$ where $[.]$ denotes the greatest integer function. The sum of these two prime numbers is