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

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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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New post 12 Jul 2019, 10:08
From the question Stem
n = 4x+3
n = 7y+1

Back solving from the answer choices

10=4*2+2=7*1+3
12 = 4*3+0 = 7+5
13 = 4*3+1 = 7+6
14=7*2+0 = 4*3+2

15 = 7*2+1 = 4*3+3

Answer 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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New post 12 Jul 2019, 10:24
1
IMO : D
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



Sol:


The numbers that are satisfying both the conditions, 15,43,71


from above we observe that there is a pattern, all the numbers are 7*2+1,7*6+1,7*10+1... so we know that adding 4 multiples of seven to these numbers

give repetitive patterns,

by this we get

2,6,10,14,18,22,26,30,34,38,42,46,50,54 th multiple of 7 and then +1

because 7*54=378 and 7*58=406, so we can take only till 54th mulptiple+1.

these number total to 14.

SO the solution is 14
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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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New post Updated on: 13 Jul 2019, 07:17
Correct answer is (E)

Originally posted by chondro48 on 12 Jul 2019, 10:27.
Last edited by chondro48 on 13 Jul 2019, 07:17, 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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New post 12 Jul 2019, 10:30
I tested the answers.

A. 10, 10/4 -> R = 2 10/7 -> R = 3
B. 12, 12/4 -> R = 0 12/7 -> R = 5
C. 13, 13/4 -> R = 1 13/7 -> R = 6
D. 14, 14/4 -> R = 2 14/7 -> R = 0
E. 15, 15/4 -> R = 3 15/7 -> R = 1

Not sure if correct, but E, works for both conditions.
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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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New post 12 Jul 2019, 10:44
1
\(n = 4a + 3\), so possible values follow the sequence \(3,7,11,15,...\)
\(n = 7b + 1\), so possible values follow the sequence \(1,8,15,...\)

by merging the common values between the two sequences, the outcome sequence is \(15,43,...\) where \(a = 15\), and \(d = 28\)
for the highest value of n < 400
\(a + (m-1)d < 400\), where m is the number of the value in the sequence
\(15 + (m-1)*28 <400\)
\(m-1 < 13\).something
\(m < 14\).something,
highest value of the integer \(m = 14\), which is the number of n within the stated constrains --> 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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New post 12 Jul 2019, 11:17
2
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?


I found this question really tricky and spent some time thinking about the concept.
What we are given:
n is a positive integer n > 0 and n < 400
n / 4 = X + 3
n / 7 = Y + 1
We need to find a number of possible "n"s.

From what we are given, n is divisable both by 4 and 7 with some remainder in both cases. Thus, the maximum number which satisfies condition n / 4 = x + 3 is n = 399 and for n / 7 = Y + 1 it is n = 394.
Since the number is divisable both by 4 and 7, we need to find out the LCM (least common multiple) of 4 and 7. LCM of 4 and 7 is the least number smallest positive integer that is divisible by both 4 and 7, which is 4 * 7 = 28.
Now let us find out how many times the number 28 is met in 400 to find the number of "n"s. Since 394 is the smallest of two possible numbers, we need to use it in the calculation: 394 / 28 = 14 and R (remainder) is 2.
Thus, the answer is 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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New post 12 Jul 2019, 12:29
2
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

Multiples of 4 having remainder 3 are

15, 43, 71 ..... 379 which is less than 400. These are following multiples of 7 ---> 2,6,10.....54. ---> this accounts to total 14 multiples.

IMO. the correct answer is (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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New post 12 Jul 2019, 12:31
1
n=4p+3 n could be 3, 7, 11, 15, 19
n=7p+1 n could be 1, 8, 15,

general formula n=mx+r; x is a divisor and r is a remainder
x would be the LCM for above two divisors (4 and 7) hence 28
r would be the first common integer in the above pattern 15

General formula based on the information above will be n=28m+15

so n could be: 15, 43, 71, 99, 127, 155, 183, 211, 239, 267, 295, 323, 351, 379 (total 14 integers)

IMO
Ans: 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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New post 12 Jul 2019, 12:41
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?

The number which we are looking for can be written as:
4a + 3
and
7 b + 1

To get the value which satisfies both equations:
4a + 3 = 7 b + 1
4 a - 7 b = -2
if we put a = 3 and b = 2 then it satisfies he equation.

4 * 7 = 28
we are looking for a number which is 28k +15 (put the value of a or b in earlier equations)

400 - 15 = 385

and

in the above equation we can put values from 0 to 13

Therefore the answer will be 14.
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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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New post 12 Jul 2019, 17:41
This can be easily solved by dividing the options with 4 and 7 and seeing what number gives the required remainders. Since answers are in ascending order I choose 13 from the middle.

13/4 gives remainder 1 but I want remainder 3 so need to increment two. 13+2=15.
Check 15/7, gives remainder 1. No need to check any other option.

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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New post 12 Jul 2019, 18:01
n=4p+3 ..(given)
n=7p+1 ... (given)

Hit and Trial Method:
Eliminate b and d because b is divisible by 4 and d is divisible by 14.

a) upon dividing a) ,which is 10, by 4 leaves a remainder of 2. eliminate it.
c) divide 13 by 4..remainder 1 ...eliminate it

Therefore, E is clear answer
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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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New post 12 Jul 2019, 18:20
1
n=1 (mod 7)
LCM(4,7)=28
Possibilities
n=1 mod 28
n=8 mod 28
n=15 mod 28
n=22 mod 28

Only 15=3 mod 4

Hence n= 15 mod 28
or n=28k+15, where k is non-negative integer

Maximum value k can take is [(399-15)/28]=13
Total values n can take= (13-0)+1=14

IMO 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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New post 12 Jul 2019, 18:31
We can answer this question by placing the answer choices one by one in the following

4n+3 and 7n+1

A. 10 = 4*2+2 = out
B. 12 = 4*3+0 = out
C. 13 = 4*3+1 = out
D. 14 = 4*3+2 = out
E. 15 = 4*3 +3 and 7*2+1 = 15

Hence the answer 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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New post 12 Jul 2019, 19:06
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

Here you know that n = 4K + 3 or 7s +1

So you can easily discard (B) and (D) since those are multiples of 4 and 7 respectively. Then you can discard 13 because the remainder when divided by 4 would be 1 and also 10 because remainder when divided by 4 would be 2.

(E) is the only possible answer.
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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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New post 12 Jul 2019, 19:34
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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

when n is divided by 4, the remainder is 3 . This can be written as n = 4k+3. Elements of this series are - 3,7,11,19,23,27,31,35,39,43,47,51.
when n is divided by 7, the remainder is 1. n = 7s + 1. Elements of this series are - 1,8,15,22,29,36,43,50

Common Element - 15,43....so on. The difference is 28.

Last element of the series can be written as \(15+28q < 400.\)
Solving this gives \(4q < 55\) or \(q <=13\).

q can take values from 0,1,2...13

Total number of n = 14

Answer - 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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New post 12 Jul 2019, 21:31
1
n = 4k + 3, for any integer k
--> n = 4k + 2 + 1 = 2(2k + 1) + 1
--> n = 2p + 1, for some positive integer p

Also, n = 7q + 1, for some integer q

The value that satisfies both conditions is n = 14m + 1, for all odd integers of m

Eg: n = 15 (m = 1), 43 (m = 3), 70 (m = 5) . . . .

Number of possible value = {15, 43, 71 , . . . . . . . . 379}
number of values = (379 - 15)/28 + 1 = 364/28 + 1
= 13 + 1
= 14

IMO Option D

Pls Hit Kudos if you like the solution
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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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New post Updated on: 13 Jul 2019, 02:34
1

D is the answer.



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?




The formula is:

a/4 = b+3 ---> a = 4b+3
If we replace b with 0, then we get 3.
If we replace b with 1, then we get 7.
If we replace b with 2, then we get 11 and so on....

Accordingly:

a/7 = b+1 ---> a = 7b+1
If we replace b with 0, then we get 1.
If we replace b with 1, then we get 8.
If we replace b with 2, then we get 15 and so on...

The first overlap is 15, the second 43. Difference is: 43-15=28.
By this we can say that, every 28 numbers we have overlap.

400/28=14.285 ~ 14
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Originally posted by GKomoku on 12 Jul 2019, 22:13.
Last edited by GKomoku on 13 Jul 2019, 02:34, 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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New post 12 Jul 2019, 22:18
1
Given:
n < 400 -> (a)
n = 4A + 3 -> (b)
n= 7B + 1 -> (c)

From (b) and (c)
n = 4A + 3 = 7B +1 -> (d)
=> 4A + 2 = 7B -> (e)

We can form an A.P. with (e) that satisfies the values of both A and B respectively.
When B = 2, A = 3, value (d) = 15
When B = 6, A = 10, value (d) = 43
When B = 10, A = 17, value (d) = 71

Thus, we see that values of n form an A.P. of form 15 + 28 D -> (f)
Where, First term = 15 and Common difference = 28
The last term of AP should be less than 400 [From (a)]
15, 43, 71…379
If 379 is the Nth term of AP then,
379 = 15 + (N – 1) 28 => N = 14

Therefore, 14 terms satisfy n.

Answer 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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New post 12 Jul 2019, 22:47
n<400

when n is divided by 4, the remainder is 3-
=> 4n+3 =>(7,11,15,19,...)

when n is divided by 7, the remainder is 1-
=> 7n+1 =>(8,15,22,29,...)

Clearly 15 is the number. Hence 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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New post 12 Jul 2019, 23:47
E it is
15/4 = 3*4+3
15/7 = 2*7+1
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Re: If n is a positive integer less than 400, what is the number of n such   [#permalink] 12 Jul 2019, 23:47

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