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# What is the 777th decimal digit of 0.345634563456……..?

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What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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17 Aug 2018, 02:44
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[Math Revolution GMAT math practice question]

What is the 777th decimal digit of 0.345634563456……..?

A. 3
B. 4
C. 5
D. 6
E. 7

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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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17 Aug 2018, 03:04
Cyclicity =4

777 when divided by 4 gives 1 as remainder. So 777th digit is A

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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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17 Aug 2018, 06:53
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the 777th decimal digit of 0.345634563456……..?

A. 3
B. 4
C. 5
D. 6
E. 7
Check carefully the digits 3456 repeats in cycle...

Thus the no of 4 digit cycles in 777th decimal is 777/4 = 194 cycles and 1 extra digit ( which is 3 )

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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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17 Aug 2018, 08:06
Top Contributor
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the 777th decimal digit of 0.345634563456……..?

A. 3
B. 4
C. 5
D. 6
E. 7

Look for a pattern...
1st decimal digit: 3
2nd decimal digit: 4
3rd decimal digit: 5
4th decimal digit: 6
5th decimal digit: 3
6th decimal digit: 4
7th decimal digit: 5
8th decimal digit: 6
9th decimal digit: 3
...etc

This pattern repeats itself every 4 digits
In other words, the pattern has a cycle of 4

So, we can see that the 4th digit is 6, the 8th digit is 6, the 12th digit is 6, etc
In other words, for every term that is a multiple of 4, the digit will be 6

Now let's find the multiple of 4 that's closest to 777
Well, 776 is a multiple of 4, so the 776th digit must be 6
This means the next digit, the 777th digit, must be 3

Cheers,
Brent
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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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17 Aug 2018, 18:20
cyclicity is 4.

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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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17 Aug 2018, 18:26
As there are 4 repeating digits in decimal so we devide 777 by 4
We get 1 remainder so in four repeating digits (3,4,5,6) first digit (3) is the answer.

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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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19 Aug 2018, 18:46
=>

The digits of 0.345634563456…. repeat with period 4.
Since the 1st, 5th, 9th, … decimal digits are 3, the n-th decimal digit is 3 if n has a remainder of 1 when it is divided by 4.
Since the 2nd, 6th, 10th, … decimal digits are 4, the n-th decimal digit is 4 if n has a remainder of 2 when it is divided by 4.
Since the 3rd, 7th, 11th, … decimal digits are 5, the n-th decimal digit is 5 if n has a remainder of 3 when it is divided by 4.
Since the 4th, 8th, 12th, … decimal digits are 6, the n-th decimal digit is 6 if n is a multiple of 4.
Since 777 has a remainder of 1 when it is divided by 4, the 777th decimal digit is 3.

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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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20 Aug 2018, 11:45
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the 777th decimal digit of 0.345634563456……..?

A. 3
B. 4
C. 5
D. 6
E. 7

The pattern is 3, 4, 5, 6. The decimal repeats every 4 digits. We see that the digit 6 is the 4th, the 8th, the 16th, etc. digit in the pattern. Thus, the digit 6 always occupies the position of a digit that is a multiple of 4. Since 776 is a multiple of 4, then the 776th digit will be a 6 and the 777th digit will be a 3.

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Re: What is the 777th decimal digit of 0.345634563456……..?  [#permalink]

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23 Aug 2018, 21:42
The question gives us a number in which digits repeats itself after 4th digit. We need to find 777th decimal digit. Lets find the remainder when 777 is divided by 4 ( as it is the cyclicity). The remainder is 1, which means the first digit of the sequence. Hence the answer is 3, option A.
Re: What is the 777th decimal digit of 0.345634563456……..?   [#permalink] 23 Aug 2018, 21:42
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