Vincen wrote:The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If p = q + 7, then which one of the following could be the value of m?
(A) 45
(B) 53
(C) 72
(D) 85
(E) 100
Like David, I would solve by PLUGGING IN THE ANSWERS.
That said, here's an algebraic approach:
When dividing by 7, the remainder can be any integer between 0 and 6, inclusive.
Thus, 0≤q≤6.
The number m yields a remainder q when divided by 7.
Thus, m is a MULTIPLE OF 7 plus q:
m =
7a + q.
The number m yields a remainder p when divided by 14.
Thus, m is a MULTIPLE OF 14 plus p:
m = 14b + p.
The prompt indicates that p = q+7.
Substituting p = q+7 into m = 14b+p, we get:
m =
14b + q + 7.
Since the expressions in blue are both equal to m, they must be equal to each other:
7a + q = 14b + q + 7
7a = 14b + 7
a = 2b + 1.
The resulting equation implies that a can be ANY ODD INTEGER.
Since a must be odd, and m = 7a + q, we get the following options for m:
m = (7*1) + q = 7 + q
m = (7*3) + q = 21 + q
m = (7*5) + q = 35 + q
m = (7*7) + q = 49 + q
And so on.
Since 0≤q≤6, the following ranges for m are possible:
m = 7+q --> m is between 7 and 13, inclusive
m = 21+q --> m is between 21 and 27, inclusive
m = 35+q --> m is between 35 and 41, inclusive
m = 49+q -->
m is between 49 and 55, inclusive.
Since 53 is within the range in green, option
B is a possible value for m.
The correct answer is
B.
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