Bunuel wrote:
Given that x is a positive integer such that 19 divided by x has a remainder of 3, what is the sum of all the possible values of x?
(A) 12
(B) 20
(C) 24
(D) 28
(E) 32
Remainder property #1: When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < DFor example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0
Since we're told that 19 divided by x leaves a remainder of 3, we know that
x > 3Remainder property #2:If N divided by D equals Q with remainder R, then N = DQ + RFor example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
Since we're not told the quotient in the question, let's say the the quotient is q.
So we can rewrite the given information as follows:
19 divided by x equals q with remainder 3This means we can conclude that 19 = xq + 3
Subtract 3 from both sides of the equation to get: 16 = xq
If 16 = xq, then we know that
x is a factor of 16, since x and q are positive integers.
We now have two great pieces of information about the value of x:
x > 3 and
x is a factor of 16The factors of 16 are: 1, 2, 4, 8, and 16
Since
x > 3, the two possible values of x are 4, 8 and 16
4 + 8 + 16 = 28
Answer: D
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