Latest TCS Aptitude Question SOLUTION: Remainder when 97! is divided by 101????

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Remainder when 97! is divided by 101????

Asked In TCS MD (10 years ago)

Unsolved Read Solution (16)

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Read Solution (16) :

PARAG KAMRA  (10 years ago)

97!/101

1*2*3*4....*97/101

rem(1/101)+rem(2/101)+rem(3/101)....+rem(97/101)

rem((1+2+3+.....+97)/101)

rem(4753/101)

=6(ans)

Is this solution Helpfull? Yes (27) | No (35)  

Pranshu  (10 years ago)

according to wilsons theorem...(n-1)!=-1(mod n)
here n is 101
so 100!=-1(mod 101)
100*99*98*97!=-1(mod 101)
97!={-1*(100*99*98)}(mod 101)
97!={-1*-1*-2*-3} (mod 101)
97!=6 (mod 101)
so answer is 6

Is this solution Helpfull? Yes (11) | No   

d g  (10 years ago)

17....using wilson's theorem

Is this solution Helpfull? Yes (6) | No (2)  

rishi pal  (10 years ago)

Wilson's Theorem is about primes, namely, for integer n>1,
(n-1)! = -1 mod n iff n is prime.

Since 101 is prime, we can use it, along with the fact that
100*99*98*97! = 100!

Then taking mod 101 and using Wilson's Theorem,
100*99*98*97! = -1 mod 101
(-1)(-2)(-3)97! = -1 mod 101

so that, multiplying both sides by -1,
6*97! = 1 mod 101

Now, we'd like to find the inverse mod 101 of 6. To do this, use the fact that
6*17 = 102 = 1 mod 101, to multiply both sides by 17, yielding

17*6*97! = 97!mod 101 = 17 mod 101

So it's 17.

Is this solution Helpfull? Yes (4) | No   

Prateek Pandey  (10 years ago)

Wilsom theorem states:
(n-1)!=-1 mod n if n is prime
Since, 101 is luckily prime
100!=-1 mod 101
now, 100*99*98*97!=-1 mod 101
inverse of 100=100 or (-1), of 99=50, of 98 is 67
so 97!=-1*100*50*67 mod 101
or 97!=-1*-1*50*67 mod 101
=3350 mod 101
=17 (ans)

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

dandaraviteja  (9 years ago)

97!/101=
97*96*............1/101
(97/101)+(96/101).......1/101
(1+2+3+4..........96+97)/101
(((1+97)/2)*97)/101=6

Is this solution Helpfull? Yes (2) | No   

Prateek Pandey  (10 years ago)

Since 101 is prime, any number in range 1 to 100 is invertible
Use Extended Euclidean algorithm given here http://en.wikipedia.org/wiki/Modular_multiplicative_inverse
A tabular method can be taken from any book on modular arithmetic.

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Pranshu  (10 years ago)

how inverse of 99 and 98 are 50 and 67 respectivly? pls explain

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Prateek Pandey  (10 years ago)

As already said, it is calculated using Extended Eucledian theorem.
It means 99*50 mod 101=1 98*67 mod 101=1

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Suyash Kunawat   (10 years ago)

@D G please explain ur answer

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MD  (10 years ago)

@ D G. wats wilson theorem?

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MD  (10 years ago)

@ D G. wats wilson theorem?

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MD  (10 years ago)

pranshu plzzzz tel me that step .... 97! vala jo first step hai,,why did u take 97!,,??????????

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Pranshu  (10 years ago)

@MD...bcoz in dis prblm v hv to calculate 97%101

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Pranshu  (10 years ago)

@prateek..please explain the concept of taking inverse which u hv taken..

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PRIYANKA AGRAWAL  (9 years ago)

remainder 97 and quotient is 0..

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