Latest Elitmus Aptitude Question SOLUTION: what is the remainder of 18!/23?

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(#M40157731) ELITMUS Numerical Ability question Keep an EYE

what is the remainder of 18!/23?

Asked In Elitmus Free soul (7 years ago)

Unsolved Read Solution (16)

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Read Solution (16) :

sooraj singh  (7 years ago)

so according to wilson's (p-2)! / p is equals to 1
now as 23 is prime so
(23-2)! /23 = 1
21! / 23 = 1
(21*20*19)*18!/23 = 1
(-2)*(-3)*(-4) * 18! /23 = 1
(-24)* 18! /23 = 1
now take the minus sign to the rhs
so 24*18! / 23 = -1
when 24 is divided by 23 remainder is 1 therefore
1* 18! / 23 = -1

as -1 doesnt make sense as a remainder so 23-1 = 22 is the remainder and the final answer .

Is this solution Helpfull? Yes (20) | No (1)  

rachna saxena  (7 years ago)

Let rem[18!/23]=r
Wilson's Theorem says rem[(p-2)!/p]=1
rem[21!/23]=1
rem[21*20*19*18!/23]=1
rem[(-2)(-3)(-4)*r/23]=1
rem[-24r/23]=1
24r=-23K+1 ... k=-1
r=23+1/24
r=1

Is this solution Helpfull? Yes (13) | No (14)  

sachin pratap singh  (7 years ago)

According to wilson theorem: { (p-1)!+1}/p=rem 0 , where p is prime no.
so;
(22!+1)/23= rem 0
so;
22!/23= rem 22 or -1
(18!*19*20*21*22)/23= rem 22
18!/23 *(-4)*(-3)*(-2)*(-1)= rem 22
18!/23*24/23=rem 22
R*1/23= rem 22
R/23=rem 22
hence R will be 45

Is this solution Helpfull? Yes (4) | No (11)  

Bikash Kumar Mahato  (7 years ago)

According to Wilson's Theorem
(p-1)!=-1(modp)
Here p is prime number(23)
(23-1)!=-1(mod23)
22!=-1(mod23)
22*21*20*19*18!=-1(mod23)
(22-23)*(21-23)(20-23)(19-23)*18!=-1(mod23)
(-1)(-2)(-3)(-4)*18!=-1(mod23)
24*18!=-1(mod23)
(24-23)*18!=-1(mod23)
18!=-1(mod23)
Hence Remainder is -1

Is this solution Helpfull? Yes (4) | No (1)  

Manisha Choudhary  (7 years ago)

Remainder is 22.

Is this solution Helpfull? Yes (4) | No (3)  

UTTAM KUMAR  (7 years ago)

According to Wilson Theorm
1+(p-1)!=0(mod p) where p is a prime no
 (p-1)!= -1 (mod p)
 (23-1)!=-1(mod 23)
 22!=-1(mod 23)
 22*21*20*19*18!=-1(mod 23)
 -1*-2*-3*-4*18!=-1(mod 23) where 22/23 give -1 rem and so on
 24*18!=-1(mod 23)
 1*18!=-1(mod 23) where 24/23 give 1 rem and so on
 So rem = -1 ie 22

Is this solution Helpfull? Yes (3) | No   

rahul shekhar  (7 years ago)

remainder is 2

Is this solution Helpfull? Yes (1) | No (3)  

sachin pratap singh  (7 years ago)

so R may be -1 or 45

Is this solution Helpfull? Yes (1) | No (5)  

himaan  (7 years ago)

Sachin can you explain to me this step

22!/23= rem 22 or -1

and
18!/23 *(-4)*(-3)*(-2)*(-1)= rem 22

how did they become -4 -3 -2 -1

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Amit Kumar Singh  (7 years ago)

18!=6.4023737e+15
and 6.4023737e+15 % 23 =11
calculate from calculator...

https://www.google.co.in/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=6.4023737e%2B15+%25+23

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rashmi negi  (7 years ago)

Ans = -1
refer this link for detailed ans:
https://www.google.co.in/search?q=what+is+the+remainder+of+18!%2F23%3F&oq=what+is+the+remainder+of+18!%2F23%3F&aqs=chrome..69i57.805j0j8&sourceid=chrome&ie=UTF-8

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Madhu  (6 years ago)

Let rem[18!/23]=r
Sincw, Wilson's Theorem says, rem[(p-2)!/p]=1
rem[21!/23]=1
rem[21*20*19*18!/23]=1
rem[(-2)(-3)(-4)*r/23]=1
rem[-24r/23]=1
24r=-23K+1 ... k=-1
r=23+1/24
r=1
Therefore, Answer is (a).

Is this solution Helpfull? Yes | No (1)  

sooraj joshi  (6 years ago)

remainder is 22

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Sayantani Maiti  (6 years ago)

As we know (p-1)!/p remainder is (-1)

Now

18!/23

=22*21*20*19*18!/22*21*20*19*23

=(23–1)!/23*22*21*20*19

=(-1)/22*21*20*19

= (-1)

So remainder is (-1) as we know there is no means of negative remainder so remainder is 23–1=22

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swati  (6 years ago)

A/wilson's theorem
rem[(p-1)!/p]=p-1
so,
rem[22!/23]=22
=> rem[22 x 21 x 20 x19/23] x rem[18!/23]=22
=>rem[24/23] x rem[18!/23]=22
=>1 x rem[18!/23]=22
hence, rem[18!/23]=22(ans)

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aakansha  (5 years ago)

(p-1)!/p=p-1 if p is prime
23 is prime
22 will be remainder

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