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What is the remainder when the number 3^1989 is divided by 7

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What is the remainder when the number 3^1989 is divided by 7 [#permalink] New post 17 May 2013, 06:43
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What is the remainder when the number 3^1989 is divided by 7?

A. 1
B. 5
C. 6
D. 4
E. 3
[Reveal] Spoiler: OA

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Last edited by Bunuel on 17 May 2013, 16:19, edited 1 time in total.
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Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink] New post 16 Apr 2014, 04:51
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seabhi wrote:
Hi Bunuel,
Can you explain this.


What is the remainder when the number 3^1989 is divided by 7?

A. 1
B. 5
C. 6
D. 4
E. 3

3^{1989}=3^{3*663}=27^{663}=(21+6)^{663}.

Now if we expand this, all terms but the last one will have 21 as a multiple and thus will be divisible by 7. The last term will be 6^{663}. So we should find the remainder when 6^{663} is divided by 7.

6^1 divided by 7 yields remainder of 6;
6^2 divided by 7 yields remainder of 1;
6^3 divided by 7 yields remainder of 6 again;
...

The remainder repeats in blocks of two: {6-1}{6-1}{6-1}... When the power is odd the remainder is 6 and when the power is even the remainder is 1. So, the remainder when 6^{663}=6^{odd} is divided by 7 is 6.

Answer: C.

Units digits, exponents, remainders problems: new-units-digits-exponents-remainders-problems-168569.html

Hope it helps.
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Re: What is the remainder??? [#permalink] New post 17 May 2013, 06:48
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3 / 7 rem = 3
3^2 = 9 / 7 rem = 2
3 ^ 3 = 27 / 7 rem = 6 or -1 --------(1)

Now, 1989/3 = 663

From (1) above,
3 ^ 1989 = (3^3) ^ 663 ;
rem = (-1) ^ 663 = -1 or 6
Ans: 6

Hope it is clear.
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Re: What is the remainder??? [#permalink] New post 17 May 2013, 09:09
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This is how I solved it:-
\frac{3^{1989}}{7}=\frac{(7-4)^{1989}}{7}
Every term in the expansion of (7-4)^{1989} would contain the number '7' except (-4)^{1989}
So it ultimately reduces to finding the the remainder when (-4)^{1989} is divided by 7.
\frac{(-4)^{1989}}{7}=\frac{(-1).(4)^{1989}}{7}=\frac{(-1).(64)^{663}}{7}
Now 64 would leave a remainder of 1 when divided by 7.
Hence the final remainder would be = -1x1=-1.
This is a negative remainder,hence for finding the actual remainder we just have to add this negative remainder to the divisor i.e. 7
Therefore, the final remainder is (-1+7)=6
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Re: What is the remainder??? [#permalink] New post 17 May 2013, 06:50
mkdureja wrote:
3 / 7 rem = 3
3^2 = 9 / 7 rem = 2
3 ^ 3 = 27 / 7 rem = 6 or -1 --------(1)
Now, 1989/3 = 663
From (1) above,
3 ^ 1989 = (3^3) ^ 663 = rem = (-1) ^ 663 = -1 or 6
Ans: 6

Hope it is clear.


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Re: What is the remainder??? [#permalink] New post 17 May 2013, 06:51
SrinathVangala wrote:
What is the remainder when the number 3^1989 is divided by 7?


A. 1
B. 5
C. 6
D. 4
E. 3


Answer would be [C] as mentioned.

3^1989 = 3^(3*663) = 27^663

The remainder left by 27/7 will be the same as the remainder left under 27^663. Hence the remainder is -1 or 6. Hope my answer is accurate!

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Re: What is the remainder??? [#permalink] New post 17 May 2013, 07:09
Can someone please explain me in detail how we arrived at the problem? I solved the Q for unit digit of the expression. Is this approach wrong? How to arrive at the solution?
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Re: What is the remainder??? [#permalink] New post 17 May 2013, 07:19
coolpintu wrote:
Can someone please explain me in detail how we arrived at the problem? I solved the Q for unit digit of the expression. Is this approach wrong? How to arrive at the solution?


Unit digit is remainder when divided by 10, what we are asked is remainder when we divide the no. by 7, so finding unit unit digit wont help you.

A rule:
If a when divided by b leaves remainder c,
then, a^x, when divided by b will leave the remainder c^x.

So, to approach the problem, we can start from raised to power 1, and go on and stop when we get 1 or -1 as remainder, then it becomes easy to solve it.
Like in this case, 3^3 leaves remainder -1 when divided by 7,
so using the above rule, we can say that 3^1989 = (3^3)^ 663 will leave remainder (-1)^663 or -1, when divided by 7.
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Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink] New post 16 Apr 2014, 04:07
Hi Bunuel,
Can you explain this.
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Re: What is the remainder when the number 3^1989 is divided by 7   [#permalink] 16 Apr 2014, 04:07
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