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# If n is a positive integer less than 400, what is the number of n such

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Manager
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 00:16
One can directly pick the numbers from the options and check. 15 is the only number when divided by 4 and 7 gives remainder of 3 and 1 respectively.
Correct Option: E
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 01:05
Number must be of the form 4A+3=7B+1.
Upon some trial and error, we get the first positive number to be 43.
Now, next numbers will be in the interval of LCM(4,7), i.e., 28. --> Next numbers 43+28=71, 99 and so on.

No. of n in the range 0 to 400 is,(Find the no. of terms in AP, assume last term is 400, we can round that off later)
400=43+28(k-1)
357/28=k-1
12.75+1=k
k=13.75
Rounding off to integer less than k, no. of n is 13.

Ans. C
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 01:37
n>0 and n< 400
n/4 leaves a remainder of 3. This means n can possibly be among the following set: {3,7,11,15}
n/7 leaves a remainder of 1 means n can also be among the following set:
{1,8,15}
Since 15 appears among the two sets and is also part of the answer choices, the answer is therefore E.

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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 02:18
1
From the question we have:
1. n=4p+3
2. n=7q+1

LCM of 4 and 7 is 28
The first common integer in these two patterns is 15

So the combined formula is n = 28z+15
we know that n should be less than 400, so z may be 13 or less (including 0)

The answer is 14 - D
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 02:37
1
Quote:
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?

A. 10
B. 12
C. 13
D. 14
E. 15

n = 4k + 3 = 7m + 1.
n + 13 = 4(k +4)= 7 (m + 2)
n+13 = multiple of 28.
therefore 28 * 15 = 420.
there 14 total n
hence option D.
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 03:12
1
N=4a+3 = 3,7,11,15,19... for a=0,1,2,....
N=7b+1 = 1,8,15,.... for b=0,1,2,...

we can combine
N=lcm(7,4)x+15 (common value)
we get N=15,43..... for x=0,1,2,....
now it is given that N<400
so we need to find x for which N<400

for x=14 N=407, hence N must have 14 possible values

note we also count x=0 as valid value hence 14 possible cases
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If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 03:18
1
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?

ans is D

the first number which fits this bill is 15. and as the divisors 4 and 7, will give other numbers in their lcm so 28 is the lcm.

so number of N =( last target no - first no) / lcm + 1

ie 400 -15 ie 385/28 +1 which is 14 . thus ans D
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If n is a positive integer less than 400, what is the number of n such  [#permalink]

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Updated on: 13 Jul 2019, 05:21
1
I used Bunuel's explanation in Remainder chapter of Math book.
Positive integer n leaves a remainder of 3 after division by 4: n=4p+3. Thus n could be: 3, 7, 11, 15, 19, ...
Positive integer n leaves a remainder of 1 after division by 7: n=7q+1. Thus n could be: 1, 8, 15, 22, ...

So, x can be 4*7=28
First integer is 15, n=28m+15. Only 13 can yield number less than 400. n=28*13+15=379. So 13 numbers + 1 (first integer 15). Answer is 14, D

Originally posted by mba2021aspirant on 13 Jul 2019, 04:02.
Last edited by mba2021aspirant on 13 Jul 2019, 05:21, edited 1 time in total.
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 04:04
1
$$n = 4p+3$$ -1
$$n = 7k+1$$ - 2
Equating 1 and 2
$$k=(4p+2)/7$$

Also, $$4p+3<400$$
=> p is less than or equal to 96.

for p=3, k=2
for p=10, k=6
difference between values of p, (d) =7
Using AP formula

$$96>3+(m-1)*7$$
$$m = 14$$
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 04:08
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?
n=4*a+3
n=7*b+1
4: 3,7,11,15,19,23,27,31,35,39,43,47,51,55,59,63,67,71
7: 1,8,15,22,29,36,43,50,57,64,71
n: 15,43,71
1. Let's focus on 7. Between 15 and 43 are only 3 numbers, and between 43 and 71 are only 3 numbers. This is a pattern, beginning from 15 every 4th number is 'n'.
2. The biggest number n=7*b+1, < 400 is 393.
So, we need to know how many '7'intervals between 15 and 393.
(393-15)/7 = 54 and quantity of numbers is 54+1 = 55 (including 15)
3. Among these 55 numbers we know from 1 point, that every 4 number is 'n'.
55/4 = 13,75 . it means that there is 13 intervals and one that doesn't complete.
So there are 13 numbers that fit condition, because in every interval 'n' is the first number.

Answ C

A. 10
B. 12
C. 13
D. 14
E. 15
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 05:38
1
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?

15 is the first such number, which can satisfy given conditions. Add 28 (LCM of 4 and 7) to 15 and we will get next number and so on.

So, numbers [15, 43, 71.........., 379] satisfy the given condition. (total 14)

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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 06:02
Best way to do this type of question is POE. Starting from c, we can clearly see 13 when divided by 4 doesn't give remainder 3. Same with D. In E when we divide 15 by 4 and 7 we get 3 and 1 resp. Hence ans is E.
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 06:18
1
IMO D.

For n to be divisible by 4 and give a remainder is 3, we get n=4m+3.
For n to be divisible by 7 and give a remainder 1, we get n=7q+1.
We need to find n such that, 4m+3 =7q+1.
The series starts with n=15,43,41,...379 -> which gives us 14 terms.
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 06:31
1
So n is a number 3 greater than a multiple of 4. It is also 1 greater than a multiple of 7.

n=4a+3
n=7b+1

n belongs to both the lists given below:

Numbers of the form (4a+3): 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75...

Numbers of the form (7b + 1): 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78...

15 , 43, 71 (and so on...) are common to both the lists

– 43 is 28 more than 15.
– 71 is 28 more than 43.
– 28 is the LCM of 4 and 7.

Now since we've identified this pattern, it's quite to easy to determine the next set of numbers in this series - upto 400, and that gives us our answer, i.e. 14 (D)
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 06:39
1
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?

A. 10
B. 12
C. 13
D. 14
E. 15

Solution:

By remainder theory we can conclude that ,

n= 4Q+ 3, & n= 7P + 1

we can apply this theory and find the various values of n by testing values of Q & P.
Since n is less than 400, n can be 15, 43, 71, 99,127, 155,183,211,239,267,295,323,351,379 since there are 14 numbers in total which satisfy the above condition ,

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If n is a positive integer less than 400, what is the number of n such  [#permalink]

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Updated on: 13 Jul 2019, 08:13
n is a positive integer less than 400.
If n is divided by 4, the remainder is 3
If n is divided by 7, the remainder is 1

What is n?

n=15,43,71,...379 all satisfy the above conditions. There are 14 nos of n in total

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Originally posted by chondro48 on 13 Jul 2019, 07:13.
Last edited by chondro48 on 13 Jul 2019, 08:13, edited 4 times in total.
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 07:23
Remainder cycle for 7 and 4 are .
0,3,2,1,0,3,2,1,.....
400/7=57
57/4=14
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 07:25
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?

0<n<400
when n is divided by 4, the remainder is 3 => n=4p+3 => n = 3, 7, 11, 15, 19...
when n is divided by 7, the remainder is 1 => n=7q+1 => n= 1, 8, 15, 22, 29...

LCM of 7 & 4 = 28
And least common value of n from both possible sets is 15.

So, n=28a+15
and we know that n<400

So, 28a+15<400
28a<385
a<13.xyz

So, the Ans should be 13. Option (C)
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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 07:37
1
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?

(1)if we divide n (0<n<400)by 4, the remainder is 3, n could be 3,7,11,15,19,23,27,31,35,39,43....
(2)if we divide n (0<n<400)by 7, the remainder is 1, n could be 1,8,15,22,29,36,43,50,57,64,71....

let's pay attention in what numbers n matches appropriately-->> they are 15,43,71....

In the second condition(which n is divided by 7...),matches is repeated every 4th number except the first one. the first one came in the 3rd position.

when n is divided by 7 (0<n<400)by 4, the remainder is 1, n could be 1,8,15,22,29,36,43,50,57,64,71...393(400 is not included)
and n comes 57 times between 0<n<400. So, we can calculate the number of n (3+4+4+4+4+4+4+4+4+4+4+4+4+4=55) which is 14 times.

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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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13 Jul 2019, 07:39
1
If n is a positive integer less than 400, what is the number of n such that when n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1?

A. 10
B. 12
C. 13
D. 14
E. 15

Solution:
According to Remainder Theory we are getting two equations:

n=4Q+3 and n=7Q+1

The value of n less than 400 for both of the cases when n divided by 4 leaving a remainder of 4 and when divided by 7 leaving a remainder 1 can b arrived by substituting various values of Q as 1,2,3,4,5,6,...…...till we get a value less than 400 in both equations.
We get the following values for both the equations:

For the first equation the various vales are=7,11,15,19,23,27,31,35,39,43,47,51,55,59,63,67,71,75,79,83,..................99,.......127,.........155,...183,....211....,239,.....267......295.....323,....351,.....379....399.
For the second equation the various value of n are=8,15,22,29,36,43,50,57,64,71,78,85,92,....99,....113,......127,.....134,.....155,.....162,...169,.......183,.....190,....211,...218,...239,......246,....260,...267,...281,....295,.....302,.....323,.....351,......358,.....365,...379,.....393.

so from above we see that the values of n for which when n divided by 4 leaves a remainder 3 and when divided by 7 leaves a remainder 1 are=15,43,71,99,127,155,183,211,239,267,295,323,351,379 .

The above numbers are 14 in count .Hence answer is D IMO.
Re: If n is a positive integer less than 400, what is the number of n such   [#permalink] 13 Jul 2019, 07:39

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