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Given that x is a positive integer such that 19 divided by x has a rem
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16 Aug 2021, 15:32
16 Aug 2021, 15:32
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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
(A) 12
(B) 20
(C) 24
(D) 28
(E) 32
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Re: Given that x is a positive integer such that 19 divided by x has a rem
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11 Nov 2021, 12:45
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Carcass 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
(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 < D
For 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 > 3
Remainder property #2:If N divided by D equals Q with remainder R, then N = DQ + R
For 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 3
This 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 16
The 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
General Discussion
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Re: Given that x is a positive integer such that 19 divided by x has a rem
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16 Aug 2021, 22:04
16 Aug 2021, 22:04
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Carcass 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
(A) 12
(B) 20
(C) 24
(D) 28
(E) 32
Number = (Quotient)(Divisor) + Remainder
19 = Qx + 3
Qx = 16
Factors of 16 are: 1, 2, 4, 8, 16
Qx = (1)(16) = (2)(8) = (4)(4) = (8)(2) = (16)(1)
\(\frac{19}{16}\) gives remainder 3
\(\frac{19}{8}\) gives remainder 3
\(\frac{19}{4}\) gives remainder 3
\(\frac{19}{2}\) gives remainder 1
\(\frac{19}{1}\) gives remainder 0
Therefore, sum of possible values of x = 16 + 8 + 4 = 28
Hence, option D
