vipulgoyal wrote:p, q, and r are positive integers. If p, q, and r are assembled into the six-digit number pqrpqr, which one of the following must be a factor of pqrpqr?
(A) 23
(B) 19
(C) 17
(D) 7
(E) none of the above
Integer PQRPQR can be rephrased as follows:
P(10�) + Q(10�) + R(10³) + P(10²) + Q(10) + R
= P(10�) + P(10²) + Q(10�) + Q(10) + R(10³) + R
= 10²P(10³ + 1) + 10Q(10³ + 1) + R(10³ + 1)
= (10³ + 1)(10²P + 10Q + R)
= (1001)(10²P + 10Q + R)
= (7*11*13)(10²P + 10Q + R).
Thus, 7 must be a factor of PQRPQR.
The correct answer is
D.
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