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HOT Competition 26 Aug/8PM: When positive integer n is divided by 14,

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Bunuel
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HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink] New post   26 Aug 2020, 20:00
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When positive integer n is divided by 14, the remainder is 3. When n is divided by 6, the remainder is 1. If List M contains all possible values of n, arranged in order of increasing numerical value, what is the tenth number in List M?

A. 409
B. 451
C. 493
D. 787
E. 871

 

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for the Heroes of Timers Competition

 

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Re: HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink] New post   26 Aug 2020, 20:08
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When positive integer n is divided by 14, the remainder is 3. When n is divided by 6, the remainder is 1. If List M contains all possible values of n, arranged in order of increasing numerical value, what is the tenth number in List M?

A. 409
B. 451
C. 493
D. 787
E. 871

We have the following 2 equations:
n= 14p + 3 (possible numbers: 3, 17, 31, 45, 59, .....)
n= 6q + 1 (possible numbers: 1, 7, 13, 19, 25, 31, ...)

31 is the 1st such number, so the 10th number will be 31 + 9* (LCM 6,14)
31 + 9*42 = 409

Answer A.
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Re: HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink] New post   26 Aug 2020, 20:12
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When positive integer n is divided by 14, the remainder is 3. When n is divided by 6, the remainder is 1. If List M contains all possible values of n, arranged in order of increasing numerical value, what is the tenth number in List M?

A. 409
B. 451
C. 493
D. 787
E. 871



value n when divided by 14 ; 3,17,31,45,59,... 73 and value of n when divided by 6 ; 1,7,13,19,25,31,37....73 we see two common terms 31 & 73 ; where first term is 31 and ∆ is 42 so 10th term would be an=a+(n-1)*d ;
a10 = 31+(10-1)*42 ; 409
option A

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Re: HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink] New post   27 Aug 2020, 01:47
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Given

    • n is divided by 14, the remainder is 3.
    • n is divided by 6, the remainder is 1.
    • List M contains all possible values of n, arranged in order of increasing numerical value.
To Find

    • The tenth number in List M?

Approach and Working Out

    • The numbers are in the form,
      o 14k + 3 and,
      o 6m + 1.
    • Hence, 14k + 3 = 6m + 1
      o 6m = 14k + 2
      o Least number satisfying this will be m = 5 and k = 2.
      o The number is 31
    • Next number will be 31 + LCM (6, 14) = 31 + 42
    • The general form will be 42x + 31.
      o The 10th number is 42 × 9 + 31
      o =409

Correct Answer: Option A
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Re: HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink] New post   27 Aug 2020, 12:33
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IMO A

n is positive integer

Dividend = (positive integer quotient)*(divisor) + remainder

=> n = 14p +3 (n divided by 14, remainder 3)---------------------eq (1)

=> n = 6q +1 (n divided by 6 , remainder 1)-----------------------eq (2)

p & q are positive integer quotient.
equating equation (1) & (2)

=>14p + 3 = 6q+1 => 6q = 2 + 14p

=> q = (1+ 7p)/ 3

for q to be positive integer, (1+7p) must be multiple of 3

=> Hence, Possible value of p will be 2, 5, 8...... as (1+7p) will be multiple of 3

=> Correspondignly value of n will be 31, 73, 115,..........(n=14p+3)

After observing series of n, we can conclude this is Arithmetic Progression

Where first term = 31 and Common difference = 42

10th Term = First Term + (10-1) * Common difference

=> 10th term of series for n = 31+ 9*42 => 31 + 378 => 409

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Re: HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink] New post   20 Sep 2021, 06:28
Bunuel wrote:
When positive integer n is divided by 14, the remainder is 3. When n is divided by 6, the remainder is 1. If List M contains all possible values of n, arranged in order of increasing numerical value, what is the tenth number in List M?

A. 409
B. 451
C. 493
D. 787
E. 871

 

This question was provided by Veritas Prep
for the Heroes of Timers Competition

 



The first 2 numbers that satisfy the possibility are 31 and 73
Therefore d=42 , a= 31

The requisite number =31 + 9*42
=> 409

Therefore IMO A
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Re: HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink] New post   25 Apr 2024, 00:13
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Re: HOT Competition 26 Aug/8PM: When positive integer n is divided by 14, [#permalink]
25 Apr 2024, 00:13
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