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When positive integer n is divided by 13, the remainder is 2. When n

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Re: When positive integer n is divided by 13, the remainder is 2. When n  [#permalink]

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New post 30 Sep 2018, 15:30
Bunuel wrote:
When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

\(1 \leqslant n \leqslant 179\,\,\,\,\,\left( {n\,\,\,\operatorname{int} } \right)\,\,\,\,\left( * \right)\)
\(n = 13M + 2\,\,\,\,,\,\,\,M\,\,\operatorname{int} \,\,\,\,\left( {\text{I}} \right)\)
\(n = 8J + 5\,\,\,\,,\,\,\,J\,\,\operatorname{int} \,\,\,\,\left( {{\text{II}}} \right)\)

\(?\,\,\,:\,\,\,n\,\,\,{\text{in}}\,\,\,\left( * \right) \cap \left( {\text{I}} \right) \cap \left( {{\text{II}}} \right)\)

\(\left( * \right)\,\, \cap \,\,\left( {\text{I}} \right)\,\,\,:\,\,\,\,\,\,1\,\, \leqslant \,\,13M + 2\,\, \leqslant \,\,179\,\,\,\,\,\mathop \Leftrightarrow \limits^{ - \,2} \,\,\,\,\, - 1 \leqslant 13M \leqslant 177\,\,\left( { = 169 + 8} \right)\)
\(- 1 \leqslant 13M \leqslant 177\,\,\left( { = 169 + 8} \right)\,\,\,\,\,\mathop \Leftrightarrow \limits^{M\,\,\operatorname{int} } \,\,\,0 \leqslant 13M \leqslant 169\,\,\,\,\,\mathop \Leftrightarrow \limits^{:\,\,13} \,\,\,\,0 \leqslant M \leqslant 13\)

\(\left( * \right)\,\, \cap \,\,\left( {{\text{II}}} \right)\,\,\,:\,\,\,\,\,\,\left\{ \begin{gathered}
\,n - 5\,\,{\text{divisible}}\,\,{\text{by}}\,\,8\,\,\,\,\, \Rightarrow \,\,\,\,\,n - 5\,\,\,\,\,{\text{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,n\,\,\,{\text{odd}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {\text{I}} \right)} \,\,\,\,\,M\,\,{\text{odd}} \hfill \\
\,n - 5\mathop = \limits^{\left( {\text{I}} \right)} 13M - 3\,\,{\text{divisible}}\,\,{\text{by}}\,\,8\,\,\,\,\, \Rightarrow \,\,\,\,\,\frac{{13M - 3}}{2}\,\,\,\,{\text{divisible}}\,\,{\text{by}}\,\,4\,\,\,\left( {***} \right)\,\,\,\,\,\, \hfill \\
\end{gathered} \right.\)

\(\left. \begin{gathered}
M = 1\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 - 3}}{2} = 5\,\,\,{\text{odd}}\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 3\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 3 - 3}}{2} = 18\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 5\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 5 - 3}}{2} = 31\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}}\,\,\,\,\,\, \hfill \\
\boxed{M = 7}\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 7 - 3}}{2} = 44\,\,\,\,\left( {***} \right)\,\,\,{\text{YES}} \hfill \\
M = 9\,\,\,\,\, \Rightarrow \,\,\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 11\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 11 - 3}}{2} = 70\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 13\,\,\,\,\, \Rightarrow \,\,\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: When positive integer n is divided by 13, the remainder is 2. When n  [#permalink]

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New post 01 Oct 2018, 03:08
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Bunuel wrote:
When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


Kudos for a correct solution.


13x+2 = 8y+5

Values of the format 13x+2 are 2, 15, 28, 41, 54, 67, 80, 93, 106...

Values of the format 8y+5 are 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93...

The first common Solution as highlighted is 93

every next solution will be at a gap that is multiple of 8 as well as a multiple of 13 i.e. at a gap that is LCM of 13 and 8 which is 104

Hence, Next solution = 93+104 = 197
Hence, Next solution = 197+104 = 301 etc.

But we have to find solutions less than 180 hence 93 is the only possible solution

Answer: Option B
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Re: When positive integer n is divided by 13, the remainder is 2. When n  [#permalink]

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New post 01 Oct 2018, 03:34
A quick lesson on remainders:

Quote:
When x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3.

When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4.

Combined, the statements above imply that when x is divided by 35 -- the LOWEST COMMON MULTIPLE OF 5 AND 7 -- there will be a constant remainder R.
Put another way, x is R more than a multiple of 35:
x = 35c + R.

To determine the value of R:
Make a list of values that satisfy the first statement:
When x is divided by 5, the remainder is 3.
x = 5a + 3 = 3, 8, 13, 18...
Make a list of values that satisfy the second statement:
When x is divided by 7, the remainder is 4.
x = 7b + 4 = 4, 11, 18...
The value of R is the SMALLEST VALUE COMMON TO BOTH LISTS:
R = 18.

Putting it all together:
x = 35c + 18.

Another example:
When x is divided by 3, the remainder is 1.
x = 3a + 1 = 1, 4, 7, 10, 13...
When x is divided by 11, the remainder is 2.
x = 11b + 2 = 2, 13...

Thus, when x is divided by 33 -- the LCM of 3 and 11 -- the remainder will be 13 (the smallest value common to both lists).
x = 33c + 13 = 13, 46, 79...


Onto the problem at hand:

Bunuel wrote:
When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


When n is divided by 13, the remainder is 2.
n = 13a + 2 = 2, 15, 28, 41, 54, 67, 80, 93...
When n is divided by 8, the remainder is 5.
n = 8b + 5 = 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93...

Thus, when n is divided by 104 -- the LCM of 13 and 8 -- the remainder will be 93 (the smallest value common to both lists).
n = 104c + 93 = 93, 197...

In the resulting list, only the value in green is less than 180.


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Re: When positive integer n is divided by 13, the remainder is 2. When n  [#permalink]

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New post 13 Oct 2019, 19:53
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Re: When positive integer n is divided by 13, the remainder is 2. When n   [#permalink] 13 Oct 2019, 19:53

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