Latest TCS Aptitude Question SOLUTION: What is the remainder when 135^77 is divided by 7 ? I am getting ANS : 6.

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(#M40036487) TCS Numerical Ability question Remainder Keep an EYE

What is the remainder when 135^77 is divided by 7 ? I am getting ANS : 6.

Asked In TCS Kataria Deepak (10 years ago)

Solved Saraswathy Read Solution (22)

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Read Solution (22) :

Saraswathy  (10 years ago)

135^1 % 7 = 2
135^2 % 7 = (2*2)%7 = 4 [since 135^2 can be written as 135^1*135^1... So 2*2]
135^3 % 7 = (4*2)%7 = 1

135^4 % 7 = (1*2)%7 = 2
135^5 % 7 = (2*2)%7 = 4
135^6 % 7 = (4*2)%7 = 1

Here the cycle is upto the power of 3
2nd cycle again started with "2" nd it goes on like 4,1
So the remainder ll be "3n+k" [since 1st cycle completed with the power of "3"]
77 => 3(25)+2
So upto 75(ie 3(25)), 25 cycles got over
Next cycle again begin with "2"
135^76 % 7 = 2
135^77 % 7 = 4
Hence reminder = 4

Ans : 4

Is this solution Helpfull? Yes (83) | No (6)  

suyash  (10 years ago)

(15^77 * 9^77)/7;
(15^77)/7 * (9^77)/7;
1 * 2^77/7;
((2^3)^25 * 2^2)/7;
1 * 4 = 4

Is this solution Helpfull? Yes (18) | No (3)  

priyanka mandal  (10 years ago)

135^77/7
now 135/7 gives reminder as 2
2^77/7, we can write 2^5.2^72/7
2^5(8^24)/7
then [2^5*(7+1)^24]/7
2^5/7
remainder is 4

Is this solution Helpfull? Yes (8) | No (1)  

surendhran  (10 years ago)

general formula is x^(n-1)%n=1
135^6%7=1 so 135^72%7=1
remaining term is 135^5%7
(15^5%7)*(9^5%7)=!
15%7=1 and 9%7=2
so (1^5%7)*(2^5%7)
32%7=4
ans is 4

Is this solution Helpfull? Yes (4) | No   

ranjan kumar  (9 years ago)

simple steps are..
135 can be written as 15*9

so (15*9)^77/7
now dividing 15 and 9 with 7 we get remainder as 1 and 2 respectively
so (1*2)^77/7

or (2)^77/7
now (2)^77 can be written as ((2)^3)^25 and rest will be 2^2
i.e ((8)^25)*(2)^2/7
or (7+1)^25*4/7
so whe we divide only 1 will remain that will give 1^25*4/7 that will give 1*4/7
so 4 will be ans..

Is this solution Helpfull? Yes (4) | No   

Swayanti Bhaumik  (9 years ago)

ans is 2
135^77/7
135 and 77 are co prime to each to other...so we can apply euler method to find the remainder
E(7)=6
77/6 the remainder is 1
thus 135^1/7 is 2....
answer is 2

Is this solution Helpfull? Yes (2) | No (4)  

miraj ansari  (9 years ago)

135^77/7
=(19*7+2)^77/7
=2^77/7
=(2^3)^25*2^2/7
=4

Is this solution Helpfull? Yes (2) | No   

Aravind  (9 years ago)

There is a shortcut method to find these type of problems

Whenever a number ends with 5 chk it's ten's digit :
Case 1:If it's a odd digit then last two digits of the number(here 135) raised to any exponent (here 77) will be 25.
Case 2:if it's an even digit then last two digits of the number will either be 25 or 75.
Here in this question we got 137^77=xxx25.
Divide xxxx25 by 7 and u will get remainder as 4.

Is this solution Helpfull? Yes (2) | No   

rituparna dutta  (9 years ago)

135/7 remainder=2
2^77/7
=((2^3)*2^74)/7
=(2^3)^24*(2^4)[as 8/7 gives remainder 1 so 8^24 gives 1.so the rest part
gives us 4/7]
4/7
=4
remainder=4

Is this solution Helpfull? Yes (1) | No   

shalu dhaka  (9 years ago)

135^77%7=(2^77)%7=((8^25)*4)%7=(1*4)%7=4

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

kumar sonal  (9 years ago)

(135)^77 =(7*19 +2)^77=2^77/7
(2^3)^25 * 2^2/7=1*4=4
hence rem is 4

Is this solution Helpfull? Yes | No   

kundan kumar  (9 years ago)

firstly divide 135 by 7 and take out the remainder. here remainder is 2.
2^77/77=(2^3)^25.2^2 . now when 8 is divided by 7 we get 1 as remainder. thus remaining term 2^2 when divided by 7 we get 4 as remainder.

Is this solution Helpfull? Yes | No   

Aaveg Barole  (9 years ago)

(15^77 * 9^77)/7;
(15^77)/7 * (9^77)/7;
1 * 2^77/7;
((2^3)^25 * 2^2)/7;
1 * 4 = 4

Is this solution Helpfull? Yes | No   

Shivali  (9 years ago)

according to formula : f(x)/x-1=f(1)
135^77 divided by 7 can be written as: (2^3)^25 * (2^2)/2^3 - 1=f(1)*2^2=4,hence the answer is 4

Is this solution Helpfull? Yes | No   

Aniruddha  (9 years ago)

answer is 4

Is this solution Helpfull? Yes | No   

sudhesh  (9 years ago)

take the last digit in 135 i.e 5 and using power cycles find out last digit i.e 5^7 is 5 now divide this last digit by 7 remainder is 2 so 2 is the answer

Is this solution Helpfull? Yes | No (2)  

indrajeet singh  (9 years ago)

ans.3

(133+2)^77 %7=2^77%7=(-1)^25 *4%7=-4%7=3

Is this solution Helpfull? Yes | No (3)  

sanatan garai  (9 years ago)

(135^77)/7)=>(2^77)/7=>[{(2^3)^25}*4]/7=>(1*4)/7=>rem=4

Is this solution Helpfull? Yes | No   

Sourav ganguly  (9 years ago)

135^1%7=2
Then according to the remainder theorem
2^77%7=(2^7)^11%7=2^7%7=128%7=2
then
2^11 %7= (2^5 * 2^6)%7=(32%7)*(64%7)=4*1=4

Is this solution Helpfull? Yes | No (1)  

abhisek banerjee  (9 years ago)

why not 6?

Is this solution Helpfull? Yes | No   

Abhisek Bag  (9 years ago)

i also getting 6

Is this solution Helpfull? Yes | No   

Subham Roy  (7 years ago)

there is a rule if p is a prime no then a^(p-1)/p gives remainder 1
so first we got 77/6=(6*12)+5
so it will become (135^6)^12 * 135^5
the term before the * sign becomes 1
no it remains 135^4
there is another rule if u have to find the remainder in the case (a*b*c*d)/N
then the answer will be multiplication of remainder in each separate division
that means (a/N)*(b/N)*(c/N)....
so if we divide 135 by 7 we get 2 as remainder. as their is 135^5 we have to multiply 2 by five times and we will get 32
again divide 32 by 7 and the final remainder is 4
ans: 4

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