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# What is the remainder when 51^203 is divided by 7?

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Math Expert
Joined: 02 Sep 2009
Posts: 60555
What is the remainder when 51^203 is divided by 7?  [#permalink]

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07 Nov 2019, 01:42
1
4
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Difficulty:

65% (hard)

Question Stats:

40% (01:19) correct 60% (01:28) wrong based on 86 sessions

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What is the remainder when $$51^{203}$$ is divided by 7?

(A) 4
(B) 2
(C) 1
(D) 6
(E) 5

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Joined: 19 Oct 2018
Posts: 1293
Location: India
Re: What is the remainder when 51^203 is divided by 7?  [#permalink]

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07 Nov 2019, 06:53
$$51^{203}$$ mod 7= $$2^{203}$$ mod 7

$$2^{203}=2^{3*67+2}= (2^3)^{67} .2^2$$

$$2^{203}$$ mod 7
= $$(2^3)^{67} .2^2$$ mod 7
= $$8^{67}*4$$ mod 7
= 4 mod 7

Bunuel wrote:
What is the remainder when $$51^{203}$$ is divided by 7?

(A) 4
(B) 2
(C) 1
(D) 6
(E) 5

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Joined: 03 Jun 2019
Posts: 1940
Location: India
GMAT 1: 690 Q50 V34
What is the remainder when 51^203 is divided by 7?  [#permalink]

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07 Nov 2019, 08:44
Bunuel wrote:
What is the remainder when $$51^{203}$$ is divided by 7?

(A) 4
(B) 2
(C) 1
(D) 6
(E) 5

Are You Up For the Challenge: 700 Level Questions

Asked: What is the remainder when $$51^{203}$$ is divided by 7?

$$2^{203} mod 7 = 2^{3*67+2} mod 7 = 8^{67}*2^2 mod 7 = 4 mod 7$$

IMO A
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What is the remainder when 51^203 is divided by 7?  [#permalink]

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07 Nov 2019, 09:34
1
1
Bunuel wrote:
What is the remainder when $$51^{203}$$ is divided by 7?

(A) 4
(B) 2
(C) 1
(D) 6
(E) 5

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$$51^{203}$$
(7*7+2)^203
last term 2^203
for terms raised to 2 ; the remainder of 7 is in pattern of 2^1 = 2; 2^2 = 4 ; 2^3 = 1
the remainder of 2^203 divided by 7 would be the same as 2^7 divided by 7 (203 is a multiple of 7)
2^7 gives remainder 1 when divided by 7 ;
IMO C

Kinshook ; could you please check my solution , I am getting answer as 1 ..
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Re: What is the remainder when 51^203 is divided by 7?  [#permalink]

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09 Nov 2019, 07:45
Imho the correct answer choice is A)
See the attached file.
Attachments

aq.PNG [ 488.35 KiB | Viewed 1287 times ]

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Re: What is the remainder when 51^203 is divided by 7?  [#permalink]

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12 Nov 2019, 20:21
Bunuel wrote:
What is the remainder when $$51^{203}$$ is divided by 7?

(A) 4
(B) 2
(C) 1
(D) 6
(E) 5

Are You Up For the Challenge: 700 Level Questions

Since 51/7 = 7 R 2, the remainder when 51^203 is divided by 7 is the same as when 2^203 is divided by 7.

Now notice that 2^203 = (2^3)^67 x 2^2 = 8^67 x 4. Since 8/7 = 1 R 1, the remainder when 8^67 is divided by 7 is the same as when 1^67 is divided by 7. Furthermore, the remainder when 8^67 x 4 is divided by 7 is the same as when 1^67 x 4 is divided by 7. Since 1^67 x 4 = 4 and 4/7 = 0 R 4, the remainder is 4.

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Re: What is the remainder when 51^203 is divided by 7?   [#permalink] 12 Nov 2019, 20:21
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