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

Question

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

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

bidskamikaze · Accepted Answer

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

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.

Archit3110 · Answer

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

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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EgmatQuantExpert · Answer

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

Superman249 · Answer

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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Crytiocanalyst · Answer

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

SamCodes · Answer

Hello there,

Thanks for your explanation!
But why do we take LCM of 6 and 7 at the end?
I mean how do we know for sure that the next number in the series can be found by adding the LCM as difference to 31?

Best Regards

Bunuel · Answer

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

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

A reminder of 3 when divided by 14 implies n = 14q + 3
A remainder of 1 when divided by 6 implies n = 6p + 1

We can derive a general formula for the total (of a type  total=dk+r , where  d  is the divisor and  r  is the remainder) based on the two formulas given: n = 14q + 3 (3, 17,  31 , 45, 59, 73, ... ) and n = 6p + 1 (1, 7, 13, 19, 25,  31 , ... ).

The divisor  d  would be the least common multiple of the two divisors 14 and 6, hence d = 42 and the remainder r would be the first common integer in the two patterns, hence r = 31.

So, the general formula based on both pieces of information is n = 42k + 31. We get: 31, 115, 157, ... So, the 10th number will be for k = 9, which is n = 42 * 9 + 31 = 409.

Answer: A.

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

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

Prep Toolkit

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