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A number is called "mystic" if it can be expressed as the sum of at le

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A number is called "mystic" if it can be expressed as the sum of at le  [#permalink]

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New post Updated on: 21 Feb 2019, 05:42
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GMATH practice exercise (Quant Class 17)

A number is called "mystic" if it can be expressed as the sum of at least three consecutive positive integers greater than 1, in which the larger addend is more than twice the smaller addend. Note that 25 (=3+4+5+6+7) and 44 (=2+3+4+5+6+7+8+9) are mystic numbers (7>2*3 and 9>2*2). Which of the following is true?

I. 124 is mystic
II. 125 is mystic
III. 126 is mystic

(A) I. only
(B) II. only
(C) III. only
(D) Exactly two of them
(E) None of them

P.S.: I have edited this post once: the previous word "parcel" was substituted by the proper word "addend".

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Originally posted by fskilnik on 19 Feb 2019, 09:17.
Last edited by fskilnik on 21 Feb 2019, 05:42, edited 1 time in total.
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Re: A number is called "mystic" if it can be expressed as the sum of at le  [#permalink]

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New post 20 Feb 2019, 12:52
fskilnik wrote:
GMATH practice exercise (Quant Class 17)

A number is called "mystic" if it can be expressed as the sum of at least three consecutive positive integers greater than 1, in which the larger parcel is more than twice the smaller parcel. Note that 25 (=3+4+5+6+7) and 44 (=2+3+4+5+6+7+8+9) are mystic numbers (7>2*3 and 9>2*2). Which of the following is true?

I. 124 is mystic
II. 125 is mystic
III. 126 is mystic

(A) I. only
(B) II. only
(C) III. only
(D) Exactly two of them
(E) None of them

\(?\,\,\,:\,\,\,M + \left( {M + 1} \right) + \left( {M + 2} \right) + \ldots + N\)

\(M \ge 2\)

\(N > 2M\,\,\,\,\,\left[ { \Rightarrow \,\,\,\,\,N > 4\,\,\,\,\,\mathop \Rightarrow \limits^{N\,\,{\mathop{\rm int}} } \,\,\,\,\,N \ge 5} \right]\)

\({{N\left( {N + 1} \right)} \over 2}\,\,\,\mathop = \limits^{{\rm{arithm}}{\rm{.}}\,{\rm{seq}}{\rm{.}}} \,\,\,\underbrace {1 + 2 + \ldots + \left( {M - 1} \right)}_{{{M\left( {M - 1} \right)} \over 2}\,\,\,\,\left[ {{\rm{arith}}{\rm{.}}\,{\rm{seq}}{\rm{.}}} \right]} + M + \left( {M + 1} \right) + \ldots + N\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {{N\left( {N + 1} \right)} \over 2} - {{M\left( {M - 1} \right)} \over 2}\)

\(2\,\, \cdot \,\,?\,\,\, = \,\,\,{N^2} + N - {M^2} + M = \left( {N - M} \right)\left( {N + M} \right) + N + M = \left( {N + M} \right)\left( {N - M + 1} \right)\)


\(\left\{ \matrix{
\,N - M + 1 \ge 3\,\,\,\,\left[ { \ge \,\,{\rm{3}}\,\,{\rm{parcels}}\,\,\left( {{\rm{stem}}} \right)\,\,{\rm{,}}\,\,{\rm{fingers}}\,\,{\rm{trick}}} \right] \hfill \cr
\,N + M > N - M + 1\,\,\,\left[ {M > 0.5} \right] \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\min \left( {N + M,N - M + 1} \right) = N - M + 1 \ge 3\,\,\,\,\left( * \right)\)


\(\left( {**} \right)\,\,\left\{ \matrix{
\,N + M\,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M = \underbrace {N + M}_{{\rm{even}}} - \underbrace {2M}_{{\rm{even}}}\,\,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M + 1\,\,{\rm{odd}} \hfill \cr
\,N + M\,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M = \underbrace {N + M}_{{\rm{odd}}} - \underbrace {2M}_{{\rm{even}}}\,\,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M + 1\,\,{\rm{even}} \hfill \cr} \right.\)



\(2\,\, \cdot \,\,?\,\,\, = \,\,\,\left( {N + M} \right)\left( {N - M + 1} \right)\,\,\,{\rm{is}}\,\,{\rm{given}}\,\,{\rm{by}}\,\,\,{\rm{even}} \cdot {\rm{odd}}\,\,\left( {**} \right),\,\,{\rm{where}}\,\,\left( * \right)\,\,\,{\rm{odd}} \ge {\rm{3}}\,{\rm{,}}\,\,{\rm{even}}\,\, \ge 4\,\,\,\,\,\left( {***} \right)\)



\({\rm{I}}{\rm{.}}\,\,\,124\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 124 = 248 = {2^3} \cdot 31\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\left\{ \matrix{
\,{\rm{odd}} = 31 \hfill \cr
\,{\rm{even}} = {2^3} \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,N + M = 31 \hfill \cr
\,N - M + 1 = {2^3} \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\, \ldots \,\,\, \Rightarrow \,\,\,\left( {N,M} \right) = \left( {19,12} \right)\)

\(124 = 12 + 13 + \ldots + 18 + 19\,\,\,{\rm{but}}\,\,{\rm{impossible}}\,\,\,\left( {N < 2M} \right)\)


\({\rm{II}}{\rm{.}}\,\,\,125\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 125 = 250 = 2 \cdot {5^3}\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,{\rm{one}}\,\,{\rm{option}}\,\,:\,\,\left\{ \matrix{
\,{\rm{odd}} = {5^2} \hfill \cr
\,{\rm{even}} = 2 \cdot 5 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,N + M = 25 \hfill \cr
\,N - M + 1 = 10 \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\, \ldots \,\,\, \Rightarrow \,\,\,\left( {N,M} \right) = \left( {17,8} \right)\)

\(125 = 8 + 9 + \ldots + 16 + 17\,\,\,{\rm{is}}\,\,{\rm{viable}}!\,\,\,\left( {N > 2M} \right)\)


\({\rm{III}}{\rm{.}}\,\,\,126\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 126 = 252 = {2^2} \cdot {3^2} \cdot 7\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,{\rm{one}}\,\,{\rm{option}}\,\,:\,\,\left\{ \matrix{
\,{\rm{odd}} = 3 \cdot 7 \hfill \cr
\,{\rm{even}} = {2^2} \cdot 3 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\mathop \ldots \limits^{{\rm{Do}}\,\,{\rm{it}}!} \,\,\,\, \Rightarrow \,\,\,\,\left( {N,M} \right) = \left( {16,5} \right)\)

\(126 = 5 + 6 + \ldots + 15 + 16\,\,\,{\rm{is}}\,\,{\rm{viable}}!\,\,\,\left( {N > 2M} \right)\)


YES, this is a very hard problem, but I believe many "pieces" of this solution are VERY useful for candidates aiming really outstanding performances!


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Our high-level "quant" preparation starts here: https://gmath.net

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Re: A number is called "mystic" if it can be expressed as the sum of at le  [#permalink]

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New post 20 Feb 2019, 13:24
Hello fakilnik, I totally agree to you but I have an easy solution for this particular problem. Trust me on this I got the same answers as yours but I did this orally.

As we can see that they have already done some work for us i.e. provided us with the sum from 2 to 9 i.e. 44.
So what I did is I kept on adding number to see if I can get near to the options.
So adding from 2 to 15 = 119
So I added from 2 to 16 = 135
Now start removing numbers from the beginning of the sequence to make nearest possible mystic number.
So I started removing 2 then 3 then 4. So the sum from 5 to 16 gives me 126, a mystic number.
Now I added 17 to the new series and started removing numbers from the start removed 5,6&7 giving me a difference of -1, so the second mystic number is 125.
We cannot proceed ahead as if we'll now remove 8 and add 18 to the list then the cond. for mystic number i.e 9*2 should be greater than max is not satisfied.

May sound like a long solution but it took me 2 mins to solve.

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Re: A number is called "mystic" if it can be expressed as the sum of at le  [#permalink]

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New post 20 Feb 2019, 13:40
varunbawa wrote:
Hello fakilnik, I totally agree to you but I have an easy solution for this particular problem. Trust me on this I got the same answers as yours but I did this orally.

As we can see that they have already done some work for us i.e. provided us with the sum from 2 to 9 i.e. 44.
So what I did is I kept on adding number to see if I can get near to the options.
So adding from 2 to 15 = 119
So I added from 2 to 16 = 135
Now start removing numbers from the beginning of the sequence to make nearest possible mystic number.
So I started removing 2 then 3 then 4. So the sum from 5 to 16 gives me 126, a mystic number.
Now I added 17 to the new series and started removing numbers from the start removed 5,6&7 giving me a difference of -1, so the second mystic number is 125.
We cannot proceed ahead as if we'll now remove 8 and add 18 to the list then the cond. for mystic number i.e 9*2 should be greater than max is not satisfied.

May sound like a long solution but it took me 2 mins to solve.

Posted from my mobile device

Hi varunbawa,

Thank you for your interest in our problem! When I created this question, I chose 44 randomly (really!), just to explain the concept.

Someone MAY find "intuitive" or "brute force" or any other not-really-technical way to express an integer that IS mystic, as a proper sum that guarantees that it is really mystic, but to prove that a number is NOT mystic (specially if it is much larger than the ones used here), well, I guess we must work on the math related, for example as I did...

Anyway, in this problem I was "generous" not to put the "All of them" alternative choice, therefore finding explicit sums for the two mystic numbers would be really enough.

Regards and success in your studies!
Fabio.
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Our high-level "quant" preparation starts here: https://gmath.net

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Re: A number is called "mystic" if it can be expressed as the sum of at le  [#permalink]

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New post 21 Feb 2019, 01:10
1
Hi fskilnik,

What's a parcel?
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Re: A number is called "mystic" if it can be expressed as the sum of at le  [#permalink]

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New post 21 Feb 2019, 05:39
smlprkh wrote:
Hi fskilnik,

What's a parcel?

Hi, smlprkh !

My mistake...sorry! It probably "came" from a free (wrong) translation of my native language Portuguese ("parcela")... The proper term is "addend" (I found googling):

Addends are numbers used in an addition problem, 2 + 3 = 5. Two and 3 are the addends, while 5 is the sum.

I will edit my question stem right after I submit this post!

Regards and success in your studies,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

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Re: A number is called "mystic" if it can be expressed as the sum of at le   [#permalink] 21 Feb 2019, 05:39
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