karthikpandian19 wrote:What is the sum of the digits of positive integer q ?
(1) The sum of the digits of q is an element of the set 226,313,447,617
(2) q = (n^3)-n for some positive integer n.
Statement 1: The sum of the digits of q is an element of the set 226,313,447, 617.
Since the sum of the digits could be 226, 313, 447, or 617, INSUFFICIENT.
Statement 2: q = (n^3)-n for some positive integer n.
q = n(n²-1) = n(n+1)(n-1).
Thus, q is the product of 3 consecutive integers: n-1, n and n+1.
Of every 3 consecutive integers, exactly one is a multiple of 3.
Thus, one of the factors of q is a multiple of 3, implying that q itself is a multiple of 3.
The sum of the digits of a multiple of 3 must also be a multiple of 3.
Thus, the sum of the digits of q must be a multiple of 3.
No way to determine the exact sum.
INSUFFICIENT.
Statements 1 and 2:
To satisfy statement 2, the sum of the digits of q must be a multiple of 3.
The sum of the digits of q must also be among the values listed in statement 1.
Of the values listed, only 447 has digits whose sum is a multiple of 3:
4+4+7 = 15.
Thus, the only multiple of 3 listed in statement 1 is 447.
Thus, the sum of the digits of q must be 447.
SUFFICIENT.
The correct answer is
C.
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