Question

what is the remainder of 9^27 is divided by 5
​

Answer 1

Answer:

The answer is 12

Solution:

In question like these, it is best to reduce the numerator to such a value that it gives either +1 or -1 from the divisor. It is also a good idea to keep on simplifying the numerator to get the answer. That way, we will be able to reach the answer quicker.

Remainder [3^27 / 25]

= Remainder [27^9 / 25]

= Remainder [2^9 / 25]

= Remainder [512 / 25] = 12

I have solved a bunch of similar questions on Quora and have compiled them in a blogpost here:

What's the remainder when 25^26 is divided by 27?

What is the remainder when 732^732 is divided by 27?

What is the remainder when 25^25 is divided by 26?

What is the remainder when 25^25 is divided by 24?

What is the remainder when 7100 is divided by 25?

*A2A*

ϕ -

ϕ(25)=ϕ(52)=25−5=20

By, ,

320=1mod25

Therefore,

327=37=33×33×3=27×27×3=2×2×3=12mod25

Hence, the remainder is 12.

In this quora answer [1], I have documented multiple techniques of finding remainder of such problems

[1]

The future of flexible thinking.

327=(33)9=(25+2)9=M(25)+29

=M(25)+512=M(25)+12

Hence the remainder of 327 when it is divided by 25 is 12.

Here M(25) stands for some multiple of 25 ( which need not be the same number) .

3^27mod25= 27^9mod25

= 2^9mod25 (as 27mod25=2)

= (2^10/2)mod25

. = (32^2/2)mod25

= (7^2/2)mod25

= (49/2)mod25

= (24/2)mod25

=12mod25

=12

Comment on my answer in case of difficulties.

What is the remainder when 2^27 is divided by 10? Request

What is the remainder of 3^247 divided by 17?

What is the remainder when 7221 divided by 25 ?

What will be the remainder when 20^23 is divided by 17?

What's the remainder when 25^25 is divided by 35?

3^27/25

Euler of 25 is 20

[ 25 = 5^2

25 (1 - 1/5) = 25 (4/5) =20 ]

It means a^20/25 = 1 remainder

3^27/25

= 3^20 * (3^3)^2 * 3/25

= 1 * 27^2 * 3/25

= 1* 2^2 * 3/25

= 12/25

=12 remainder

I don't know why my answer is collapsing thats why im typing this uselsss thing

Become a certified user experience designer today! Apply now!

To check if any number is divisible by 25 we just need to check if the number formed with last two digits is divisible by 25.

Now we know if the number is divisible by 25. Since the 3 is already less than 25 we are not going to need that for this particular question.

Now trying to get all the remainders for 1,10 powers of 3, the moment we get 1 or -1 we shall stop.

3,9,2,6,-7.. wait this -7 we would give us 49 or -49 and we 1 near to 50.

3^5 -> -7

3^10 -> -1 <- problem solved, now we just need to do some basic things.

3^20->1

3^(20*something + something else)

3^(20*1+7)

3^7, now since till 5 we alread

Let a be the remainder when 3^27 is divided by 25.

Thus 3^27=a mod 25

(27)^9=a mod 25

2^9=a mod 25

2^5*2^4=a mod 25

7*16 =a mod 25

112=a mod 25

=>a=12.

Hence 12 is the remainder when 3^27 is divided by 25.

33≡2(mod25)

327=33×9≡29(mod25)

29=512≡1(mod25)

Hence the remainder is 12 .

3^3 mod 25= 27 mod 25= 2

=> 3^27 mod 25= 2^9 mod 25

2^4 mod 25= 16 mod 25= -9

2^5 mod 25= 7

=> 2^9 mod 25= -63 mod 25= (-75+12) mod 25= 12 (Answer)

Arun Iyer's answer is much more elegant though

3^27

=(3³)^9

=(25+2)^9

≡2^9 mod 25

≡2×2^8

≡2(250+6)

≡12mod 25

What's the remainder when 25^26 is divided by 27?

What is the remainder when 732^732 is divided by 27?

What is the remainder when 25^25 is divided by 26?

What is the remainder when 25^25 is divided by 24?

What is the remainder when 7100 is divided by 25?

What is the remainder when 2^27 is divided by 10? Request

What is the remainder of 3^247 divided by 17?

What is the remainder when 7221 divided by 25 ?

What will be the remainder when 20^23 is divided by 17?

What's the remainder when 25^25 is divided by 35?

What is the remainder when 2^33 is divided by 27?

What will be the remainder when 25 is divided by 1331?

What is the remainder when 1! +2! + 3! + … + 25! Is divided by 13?

What is the remainder when 25^2015 is divided by 18?

What is the remainder when -27 is divided by 5?

Answer 2

Answer:

The remainder when  $9^{27}$ is divisible by 5  = 4

Step-by-step explanation:

To find,

The remainder when $9^{27}$ is divided by 5

Solution:

We have

9¹ = 9

9² = 81

9³ = 729

9⁴ = 6561

9⁵ = 59049

9⁶ = 531441

If we analyze the powers of 9 we can conclude that all the odd powers of 9 end with 9 and all the even powers of 9 end with 1.

Since the given numbers is $9^{27}$, here the power is an odd number

Hence, the number  $9^{27}$ends with the digit 9.

When we divide a number with unit digit 9 by 5, we get the reminder 4

∴ The remainder when  $9^{27}$ is divisible by 5  = 4

#SPJ3