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26 May 2017, 04:48
Kudos
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For a positive integer \(n, x=10^{n}-27\). What is the remainder when \(x\) is divided by 9?
A. 0
B. 1
C. 2
D. 3
E. 4
A. 0
B. 1
C. 2
D. 3
E. 4
26 May 2017, 04:49
Kudos
Bookmarks
Official Solution:
For a positive integer \(n, x=10^{n}-27\). What is the remainder when \(x\) is divided by 9?
A. 0
B. 1
C. 2
D. 3
E. 4
The remainder of an integer n divided by either 3 or 9 is the same as the remainder of the sum of all the digits of integer n divided by 3 or 9.
With \(n = 2, x = 10^{2}-27 = 73\), so the number of 9s is 0.
With \(n=3, x = 10^{3}-27 = 973\), so the number of 9s is 1.
With \(n = 4, x = 10^{4}-27 = 9973\), so the number of 9s is 2.
...
Since the remainder of integer n divided by 9 is the same as the remainder of the sum of all the digits of x divided by 9, the sum of all the digits of x is \(9(n-2)+7+3 = 9(n-2)+10\). When \(9(n-2) + 10\) is divided by 9, since \(9(n-2)\) is divisible by 9, you just need to divide the number 10 by 9. The remainder of 10 divided by 9 is 1, so x divided by 9 has the remainder of 1. The answer is B.
Answer: B
For a positive integer \(n, x=10^{n}-27\). What is the remainder when \(x\) is divided by 9?
A. 0
B. 1
C. 2
D. 3
E. 4
The remainder of an integer n divided by either 3 or 9 is the same as the remainder of the sum of all the digits of integer n divided by 3 or 9.
With \(n = 2, x = 10^{2}-27 = 73\), so the number of 9s is 0.
With \(n=3, x = 10^{3}-27 = 973\), so the number of 9s is 1.
With \(n = 4, x = 10^{4}-27 = 9973\), so the number of 9s is 2.
...
Since the remainder of integer n divided by 9 is the same as the remainder of the sum of all the digits of x divided by 9, the sum of all the digits of x is \(9(n-2)+7+3 = 9(n-2)+10\). When \(9(n-2) + 10\) is divided by 9, since \(9(n-2)\) is divisible by 9, you just need to divide the number 10 by 9. The remainder of 10 divided by 9 is 1, so x divided by 9 has the remainder of 1. The answer is B.
Answer: B
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