M7MBA wrote:If n is a 27-digit positive integer, all of whose digits are the same, which of the following must be true?
I. n is divisible by 3
II. n is divisible by 9
III. n is divisible by 27
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III
An integer is a multiple of 9 if its digit sum is a multiple of 9.
Test the smallest possible case:
Let n = 111, 111,111...111,111,111, where all 27 digits are 1.
Since there are 27 digits of 1, the sum of the digits = 27*1 = 27.
Since the digit sum is a multiple of 9, n is divisible by 9.
Since n is divisible by 9, it must also be divisible by 3.
Since 111/3 = 037, dividing n by 3 will yield the following:
37, 037, 037...037, 037, 037.
Here, there will be 9 sets of 037, with the result that the integer in blue will have the following digit sum:
9*37.
Since the digit sum is a multiple of 9, the integer in blue is divisible by 9.
Thus:
n can be divided by 3 and then by 9, implying that n is divisible by 27.
Since the smallest option for n is divisible by 3, 9 and 27, any larger option for n will also be divisible by 3, 9 and 27.
The correct answer is
E.
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