a, b, and c are positive integers. If a, b, and c are assembled into the six-digit number abcabc, which one of the following must be a factor of abcabc?
  • a)
    16
  • b)
    13
  • c)
    5
  • d)
    3
  • e)
    none of the above
Correct answer is option 'B'. Can you explain this answer?
Related Test: GMAT Full Mock Test- 1

GMAT Question

This question looks daunting, but we can tackle it by thinking about the place values of the unknowns. If we had a three-digit number abc, we could express it as 100a + 10b + c
(think of an example, say 375: 100(3) + 10(7) + 5). Thus, each additional digit increases
the place value tenfold.
If we have abcabc, we can express it as follows:
100000a + 10000b + 1000c + 100a + 10b + c
If we combine like terms, we get the following:
100100a + 10010b + 1001c
At this point, we can spot a pattern in the terms: each term is a multiple of 1001. On the
GMAT, such patterns are not accidental. If we factor 1001 from each term, the
expression can be simplified as follows:
1001(100a + 10b + c) or 1001(abc).
Thus, abcabc is the product of 1001 and abc, and will have all the factors of both. Since
we don't know the value of abc, we cannot know what its factors are. But we can see
whether one of the answer choices is a factor of 1001, which would make it a factor of
abcabc.
1001 is not even, so 16 is not a factor. 1001 doesn't end in 0 or 5, so 5 is not a factor.
The sum of the digits in 1001 is not a multiple of 3, so 3 is not a factor. It's difficult to
know whether 13 is a factor without performing the division: 1001/13 = 77. Since 13
divides into 1001 without a remainder, it is a factor of 1001 and thus a factor of abcabc.
The correct answer is B.

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