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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……..?

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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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Max@Math Revolution 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
Determine the number of times the four-digit pattern "3456" is repeated.
777 ÷ 4 = 194 R1, so the four-digit pattern occurs 194 times. The remainder of 1 indicates that the 777th digit is the first digit in the next repetition of the pattern.
The correct answer is choice A.

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Max@Math Revolution 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

Answer: A

Cheers,
Brent

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=>
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.
Therefore, the answer is A.

Answer: A

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Max@Math Revolution 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.

Answer: A

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