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What is the remainder when (25^99 x 4^99)^99 is divided by 11?

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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 11:25
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What is the remainder when \((25^{99}*4^{99})^{99}\) is divided by 11?

\([(25^{99}]*[4^{99})]^{99}= [100^{99}]^{99}=[10^100]^{99}=[10^{9900}]\)

This is going to be followed by a lot of 0's
we need to see what happens when we break it down..

10 remainder =10
100/11 remainder = 1
1000/11 remainder = 10
10000/11 remainder = 1

which means for
\(10^{odd number}\) remainder =10
\(10^{Even number}\) remainder =1

[10^9900] => even number, so the remainder = 1

Answer A
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 12:31
1
What is the remainder when \((25^{99}*4^{99})^{99}\) is divided by 11?

10/11 --> remainder = 10
100/11 --> remainder = 1
1000/11 --> remainder = 10
10000/11 --> remainder = 1

Therefore, if we have odd no. of 0s then remainder = 10
and if we have even no. of 0s then remainder = 1

Now \((25^{99}*4^{99})^{99}\) = (100^99)^99
If we raise 100 by any number, we will always end up with even no. of 0s.
Hence, when divided by 11, remainder = 1


A. 1
B. 3
C. 7
D. 9
E. 10

Answer Choice: A
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 12:31
1
Quote:
What is the remainder when \((25^{99}*4^{99})^{99}\) is divided by 11?

A. 1
B. 3
C. 7
D. 9
E. 10


\(25^{99}\)=\(5^{2*99}\)
\(4^{99}\)=\(2^{2*99}\)
therefore,
\(((5*2)^{2*99})^99\)

=\(10 ^{2*99*99}\)
Now \(10^2\) has remainder 1 when divided by 11
so R(\(\frac{[m](10^2)^{99*99}\) }{ 11}[/m])
= \(1^ (99*99)\)
= 1
Hence option A.
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 12:39
1
N=\((25^{99}*4^{99})^{99}\) =\((100^{99})^{99}\)
When 100 is divided by 11, remainder is 1. Therefore, when N is divided by 11 remainder is \((1^{99})^{99}\)=1

Ans: A
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 13:02
1
What is the remainder when (25^99 x 4^99)^99 is divided by 11?

A. 1
B. 3
C. 7
D. 9
E. 10

The only little step we need to take is to recall that (25^99) * (4^99) = 100^99
And that's it.
We can easily find out that 100 / 11 will give us the reminder 1.
The same is for (100^2)/11 and for (100^n)/11

So the answer is A
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What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post Updated on: 20 Jul 2019, 05:01
The remainder when \(100^{99*99}\) is divided by \(11\) is... \(10\). Sorry for the abrupt spoiler. Let's figure out why it is \(10\).

The reminder depends on the number of \(0\)s \(10^{x}\) has. For example:

\(\frac{10}{11}\) - remainder is \(10\)

\(\frac{100}{11}\) - remainder is \(1\)

\(\frac{1000}{11}\) - remainder is \(10\)

\(\frac{10000}{11}\) - remainder is \(1\)

We can see that if the number of \(0\)s following \(10^{x}\) is odd then the remainder, if divided by \(11\), is \(10\). If it is even, then the remainder is \(1\).

\(10^{2*99*99}\) will have even number of \(0\)s because \(2*99*99\) is an even number. Hence, \(\frac{{10^{2*99*99}}}{11}\) will have remainder a \(1\).

Hence A
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Originally posted by JonShukhrat on 19 Jul 2019, 14:10.
Last edited by JonShukhrat on 20 Jul 2019, 05:01, edited 1 time in total.
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 14:46
1
Answer is A

Question is asking what is the remainder when (25^99* 4^99)^99 is divided by 11?

(25^99* 4^99)^99
= (100^99)^99
When we divide 11 with 100 with any power the remainder will always be 1, for example: 100^1/11 Remainder is 1,
100^2/11 Remainder is 1, 100^3/11 Remainder is 1... Therefore we do not need to consider the huge power that 100 has in this question because no matter what power 100 has, when it is divided by 11 it will always provide us with the remainder 1.

Answer is A
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 15:25
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What is the remainder when (25^99•4^99)^99 is divided by 11?

((100)^99)^99/11 —r—> since remainders can be multiplied ,we treat it as dividend —-> ((1^99)^99)/11 —r—-> 1/11

Hence remainder is 1
Answer A

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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 18:31
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What is the remainder when (25^99 ∗ 4^99 )^99 is divided by 11

This means (25*4)^99^99

And 100 / 11 will always be 1.

Therefore 100 Raised to any number is 1

Answer : 1

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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 19:49
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Question:- 100^9801/11

(100^10)980/11 gives remainder 1.

So 100/11 gives remainder 1.

Hence Answer A
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 20:06
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Remainder 1
Expanding the value would give 10^even power, when it is even the remainder would be 1 and when it is odd the remainder would be 10


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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 20:13
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\((25^{99}∗4^{99})^{99}\)
--> \((100^{99})^{99}\)
--> \((99 + 1)^{99*99}\)
--> \((11K + 1)\) ALWAYS for some positive integer K

So, Remainder when 11K + 1 divided by 11 = 1

IMO Option A

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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 20:48
What is the remainder when (25^99∗4^99)^99 is divided by 11?

(25^99∗4^99)^99

(100^99)^99
100^(99*99)

100/11 leaves remainder 1
1000/11 leaves remainder 10
10000/11 leaves remainder 1
100000/11 leaves remainder 10

10^odd leaves remainder 1
10^even leaves remainder 10

So, 100^(99*99) leaves remainder 10.
Option E.
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 20:58
What is the remainder when\((25^{99}∗4^{99})^{99}\) is divided by 11?

\(((25*4)^{99})^{99}\)
\((100^{99})^{99}\)


Try 10/11 Remainder = 10
100/11 Remainder = 1
1000/11 = 10
10000/11 = 1
notice the cycle 10's -> for odd number of power of 10 remainder is 10 and for even number of power of 10, remainder is 1

in above - \((100^{99})^{99}\) - 99 * 99 = unit digit is 1 -> so would be a odd number , hence the remainder would be 10

E is the right answer!
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What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 21:10
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(25^99 x 4^99)^99 = (5^2*99 x 2^2*99)^99 = (5 * 2)^2*99*99 = 10^(2*99*99)

Now, lets look at the pattern for remainder when 10^x is divided by 11:

Remainder of 10^1 /11 is 10
Remainder of 10^2 /11 is 1
Remainder of 10^3 /11 is 10
so on..

Okay, so remainder of 10^(odd) /11 is 10 and remainder of 10^(even)/11 is 1.

Since even*odd*odd (2*99*99) is even, the remainder of (10^(2*99*99))/11 is 1.
Answer is A.
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What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post Updated on: 21 Jul 2019, 01:02
(25^99*4^99)^99 ; 100^99*99 ; remainder would be 1 when divided by 11
IMO A

What is the remainder when (2599∗499)99(2599∗499)99 is divided by 11?

A. 1
B. 3
C. 7
D. 9
E. 10
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Originally posted by Archit3110 on 19 Jul 2019, 21:13.
Last edited by Archit3110 on 21 Jul 2019, 01:02, edited 1 time in total.
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 21:20
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simplifying gives the expression as (100^99)^99

Divide by 11

{[(99+1)^99]^99}/11

R = [(0+1)^99]^99

Remainder = 1

Hence, A is the Answer.
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 21:35
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(25^99 * 4^99) = 100^99
100 when divided by 11, leaves a remainder of 1. Hence, 100^99 when divided by 11 also leaves a remainder of 1.
And so does (100^99)^99
Hence, the answer is (A).
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What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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New post 19 Jul 2019, 21:45
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The total number resolves to - \((100^{99})^{99}\)

Now, note that\(\frac{100^1}{11} = Remainder = 1.\)
Similarly, \(\frac{100^2}{11} = \frac{10000}{11} = Remainder = 1.\)

For any power of 100, remainder will be 1.

Hence Option - A
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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

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Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?   [#permalink] 19 Jul 2019, 22:01

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