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Senior Manager  Joined: 21 Oct 2013
Posts: 411
When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 72% (02:06) correct 28% (02:07) wrong based on 319 sessions

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When m is divided by 9, the remainder is 2. When m is divided by 13, the remainder is 8. If 1 < m < 200, what is the greatest possible value of m?

A. 47
B. 65
C. 103
D. 117
E. 164

OE:
Nearest multiple of 9 to 164 = 18×9=162, leaving a remainder of 2.
Nearest multiple of 13 to 164 = 12×13 = 156, leaving a remainder of 8.
Thus 164 fits both conditions, and the greatest answer choice

Originally posted by goodyear2013 on 24 Mar 2014, 03:23.
Last edited by Bunuel on 24 Mar 2014, 03:45, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58446
Re: When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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goodyear2013 wrote:
When m is divided by 9, the remainder is 2. When m is divided by 13, the remainder is 8. If 1 < m < 200, what is the greatest possible value of m?

A. 47
B. 65
C. 103
D. 117
E. 164

OE:
Nearest multiple of 9 to 164 = 18×9=162, leaving a remainder of 2.
Nearest multiple of 13 to 164 = 12×13 = 156, leaving a remainder of 8.
Thus 164 fits both conditions, and the greatest answer choice

When m is divided by 9, the remainder is 2 --> $$m=9q+2$$ --> m can be 2, 11, 20, 29, 38, 47, ...
When m is divided by 13, the remainder is 8 --> $$m=13p+8$$ --> m can be 8, 21, 34, 47, 60, 73, ...

There is a way to derive general formula for m (of a type $$m=dx+r$$, where d is divisor and r is a remainder) based on above two statements:

Divisor d would be the least common multiple of above two divisors 9 and 13, hence $$d=117$$ and the remainder r would be the first common integer in above two patterns, hence $$r=47$$.

Therefore general formula based on both statements is $$m=117x+47$$ --> n can be 47, 164, 211, ... Since m must be less than 200, then the greatest possible value of m is 164.

More about deriving general formula for such problems at: manhattan-remainder-problem-93752.html#p721341

Hope it helps.
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Re: When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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2
Looking at the choices, options B & D can be eliminated as they are divisible by 13

Starting from the highest number 164;

164 = 162 + 2 (162 divisible by 9)

164 = 156 + 8 (156 divisible by 13)

it fits the result

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Re: When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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An obvious approach would be to start with the largest number
Fortunately it satisfies both the remainder criteria..E it is
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Re: When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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goodyear2013 wrote:
When m is divided by 9, the remainder is 2. When m is divided by 13, the remainder is 8. If 1 < m < 200, what is the greatest possible value of m?

A. 47
B. 65
C. 103
D. 117
E. 164

I would probably go with Bunuel's solution, but here's another approach:

There's a nice rule that says "If a number is divisible by 9, the sum of its digits will be divisible by 9"
For example, we know that 551008 is divisible by 9, because the sum of its digits equals 18, and 18 is divisible by 9

We're told that when m is divided by 9, the remainder is 2.
So, my is TWO GREATER than some multiple of 9.

So, for a possible value of m to meet this condition, all we need to do is subtract 2 from that value and see if it is divisible by 9.
Let's check the answer choices:

A. 47 - 2 = 45 This is divisible by 9, so m MIGHT equal 47
B. 65 - 2 = 63 This is divisible by 9, so m MIGHT equal 65
C. 103 - 2 = 101 This is NOT divisible by 9, so ELIMINATE C
D. 117 - 2 = 115 This is NOT divisible by 9, so ELIMINATE D
E. 164 - 2 = 162 This is divisible by 9, so m MIGHT equal 164

Let's start with the largest answer choice (E) and see if it meets the second condition (When m is divided by 13, the remainder is 8.)
164 divided by 13 = 12 with remainder 8
DONE

Cheers,
Brent
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Re: When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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This question can be solved just going by option ,

greatest value , start from E and you have ur answer ...easiest and sound way
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When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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goodyear2013 wrote:
When m is divided by 9, the remainder is 2. When m is divided by 13, the remainder is 8. If 1 < m < 200, what is the greatest possible value of m?

A. 47
B. 65
C. 103
D. 117
E. 164

assume difference of 1 between m/9 and m/13 quotients
(m-2)/9-(m-8)/13=1
4m=71 assume difference of 2
4m=71+117=188 m=47
47+9*13=164
E
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Re: When m is divided by 9, the remainder is 2. When m is divide  [#permalink]

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_________________ Re: When m is divided by 9, the remainder is 2. When m is divide   [#permalink] 06 Sep 2018, 01:37
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