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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   75% (hard)

Question Stats: 41% (01:33) correct 59% (01:44) wrong based on 29 sessions

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What is the remainder when you divide $$2^{200}$$ by 7?

A. 1
B. 2
C. 3
D. 4
E. 5

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Math Expert V
Joined: 02 Sep 2009
Posts: 58427

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Official Solution:

What is the remainder when you divide $$2^{200}$$ by 7?

A. 1
B. 2
C. 3
D. 4
E. 5

We cannot compute $$2^{200}$$ in anywhere near the time allotted, so we should look for a pattern in much simpler problems that we can scale up to $$2^{200}$$.

The simpler problems we should solve are these:

What is the remainder when you divide 2 by 7?

What is the remainder when you divide $$2^2$$ by 7?

What is the remainder when you divide $$2^3$$ by 7?

What is the remainder when you divide $$2^4$$ by 7?

... and so on with the powers of 2.

2 divided by 7 leaves remainder 2.

$$2^2$$ (which equals 4) divided by 7 leaves remainder 4.

$$2^3$$ (which equals 8) divided by 7 leaves remainder 1.

$$2^4$$ (which equals 16) divided by 7 leaves remainder 2.

$$2^5$$ (which equals 32) divided by 7 leaves remainder 4.

$$2^6$$ (which equals 64) divided by 7 leaves remainder 1.

We should stop as soon as we notice that the cycle will repeat itself forever in this pattern: [2, 4, 1]. Every third remainder is the same. (From here on out, "remainder" always means "remainder after we divide by 7.") Since every third remainder is the same, we should look at the remainder when the power is a multiple of 3. The remainders of 23 and 26 are 1. Thus, the remainder of 2 raised to a power that is any multiple of 3 is 1.

Now, 200 is not a multiple of 3, but we can look for a multiple of 3 near 200. 201 is a multiple of 3 (its digits add to 3), so $$2^{201}$$ has a remainder of 1. Finally, we notice that the remainder of $$2^{200}$$ must be one position earlier in the cycle than the remainder of $$2^{201}$$. Since the cycle is [2, 4, 1], the remainder of $$2^{200}$$ is 4.

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1
2^5 /7 gives 4 as remainder. Hence, power 200, which is multiple of 5, will give remainder as 4, as well
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Joined: 03 Jul 2018
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Saps

That's a wrong deduction. Since 200 is a multiple of 2^1 , 2^2, 2^3 ....etc .

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Bunuel,

Can you please explain the last part of the answer - "Now, 200 is not a multiple of 3, but we can look for a multiple of 3 near 200. 201 is a multiple of 3 (its digits add to 3), so 22012201 has a remainder of 1. Finally, we notice that the remainder of 22002200 must be one position earlier in the cycle than the remainder of 22012201. Since the cycle is [2, 4, 1], the remainder of 22002200 is 4."

I got stuck as I chose 2^198 as the closest no divisible by 3, then 200 is two positions after in the cycle (2,4,1) and so the remainder is 1 ??!

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Math Expert V
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Posts: 58427

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Geetshri wrote:
Bunuel,

Can you please explain the last part of the answer - "Now, 200 is not a multiple of 3, but we can look for a multiple of 3 near 200. 201 is a multiple of 3 (its digits add to 3), so 22012201 has a remainder of 1. Finally, we notice that the remainder of 22002200 must be one position earlier in the cycle than the remainder of 22012201. Since the cycle is [2, 4, 1], the remainder of 22002200 is 4."

I got stuck as I chose 2^198 as the closest no divisible by 3, then 200 is two positions after in the cycle (2,4,1) and so the remainder is 1 ??!

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Check here: https://gmatclub.com/forum/what-is-the- ... 40821.html
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I agree with explanation.
Intern  Joined: 28 Nov 2018
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how can 2 divided by 7 = 2 remainder?
Math Expert V
Joined: 02 Sep 2009
Posts: 58427

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7sharma wrote:
how can 2 divided by 7 = 2 remainder?

Let me ask you a question: how many leftover apples would you have if you had 1 apple and wanted to distribute in 9 baskets evenly? Each basket would get 0 apples and 1 apple would be leftover (remainder).

When a divisor is more than dividend, then the remainder equals to the dividend, for example:
3 divided by 4 yields the reminder of 3: $$3=4*0+3$$;
9 divided by 14 yields the reminder of 9: $$9=14*0+9$$;
1 divided by 9 yields the reminder of 1: $$1=9*0+1$$.

6. Remainders

For more:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
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