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Number theory problem 1:

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Number theory problem 1:

by \'manpreet singh » Wed Sep 04, 2013 9:25 pm
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

OA=B
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by vipulgoyal » Wed Sep 04, 2013 9:39 pm
I think plug in will be the fastest way to solve this

Start with C
if m = 72, then 72/14 = 70 with a remainder of 2. This is already out because we know that P is at least 7 (if q = 0.)same with 85 reminder 1, 100 reminder 2,
check only for those with reminder greater then 7
Now try choice B. If m = 53, then 53/14 = 42 with a remainder of 11, and 53/7 = 7
with a remainder of 4. So we have p = 11 (from above) and q = 4,
we're given that (p = q + 7), therefore, the correct answer is (B)
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by ganeshrkamath » Thu Sep 05, 2013 12:05 am
'manpreet singh 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

OA=B
m = 14a + p_________(1)
m = 7b + q_________(2)

From (1)
m = 7(2a) + q + 7
m = 7(2a + 1) + q_________(3)

Comparing (2) and (3)
b = 2a + 1

So the correct option, when divided by 7, should give an odd quotient.
45 = 7*6 + 3
53 = 7*7 + 4
72 = 7*10 + 2
85 = 7*12 + 1
100 = 7*14 + 2

Choose B

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by GMATGuruNY » Thu Sep 05, 2013 10:00 am
'manpreet singh 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

OA=B
We can plug in the answers, which represent the value of m.

When the answer choices are divided by 14, we get:
A: 45/14 = 3 R3, implying that p=3.
B: 53/14 = 3 R11, implying that p=11.
C: 72/14 = 5 R2, implying that p=2.
D: 85/14 = 6 R1, implying that p=1.
E: 100/14 = 7 R2, implying that p=2.
Since p = q+7, p≥7.
Eliminate A, C, D and E.

The correct answer is B.
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