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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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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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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.
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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, 07:53
1
I don't have a proper approach or formula. I started by counting the multiples of 7 and +1. That will give remainder as 1 when divided by 7
8 --- remainder as 1 when divided by 7, rem=0 when divided by 4
15--- remainder as 1 when divided by 7, rem=3 when divided by 4
22 --- remainder as 1 when divided by 7, rem=2 when divided by 4
29 --- remainder as 1 when divided by 7, rem=1 when divided by 4
36 --- remainder as 1 when divided by 7, rem=0 when divided by 4
43 --- remainder as 1 when divided by 7, rem=3 when divided by 4

We have a pattern starting from 15 and increasing the number by 28 we have the number that satisfies this condition.
Last number in the pattern less than 400 is 379.

Total number of 'n' that satisfies the condition 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, 07:54
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

Let's break down the question;-

N= 3 or 7 or 11 or 15 keep adding 4
N= 1 or 8 or 15 or 22 or 29 keep adding 7

The key to master this question is pattern recognition So N which falls in both the criteria is 15, 43, 71 difference between each number is 28, so keep adding till 400

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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, 07:56
1
n=4a+3
Substituting whole number values for a, we get n=3,7,11,15,19,23,27,31,35,39,43...
n=7b+1
Substituting whole number values for a, we get n=1,8,15,22,29,36,43,..
We see that the first common number is 15; then it's 43.
So the difference between 2 consecutive numbers is 28. So there are 14 such numbers upto 400.

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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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12 Jul 2019, 08:09
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

15 is the number which leaves 3 and 1 with 4 and 7 respectively.

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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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12 Jul 2019, 08:11
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?

given:
n = 4*p + 3=7*q + 1, where p & q are integers

A. 10 --> 10 = 4*2 + 2
B. 12 --> 12 = 4*3 + 0
C. 13 --> 13 = 4*3+1
D. 14 --> 14 = 4*3+2
E. 15 --> correct: 15 = 4*3 + 3 = 7*2+1
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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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12 Jul 2019, 08:19
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

The values that satisfy both the condition are 15,43,71.
So if we write the equation the number has to satisfy 4x+3 and 7y+1. The values of y which satisfy both the equation are y=2,y=6,y=10,14,18 so we see that it jumps by 4. so last number is going to be y=42 which when put into equation number turns out to be 54.
hence total numbers to satisfy the condition is 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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12 Jul 2019, 08:28
C

1st condition: n = 4a+3 = 3,7,11,15,19,23...
2nd condition: n=7b+ 1 = 1,8,15,22,...

So, first common integer in both series is 15. GCD of 4 and 7 is 28.

Therefore, n = 28*x + 15. Now, since n has to be less than 400, we have: 28*x + 15 < 4000 => x < 13.5..
So maximum value of x is 13.
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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: 15 Jul 2019, 11:11
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

0 < n positive < 400
when n/4 remainder is 3: n={3 7 11 [15] 19 23…[43]…}
when n/7 remainder is 1: n={1 8 [15] 22 29 36…[43]…}
the pattern of n such numbers that fit the condition has a common difference of 28 [=43-15]
thus we need to find the last term with a common difference of 28 from 0 to 400,
with a first term of 15 and a last term of 15+28(x-1)<400:
15+28(x-1)<400,…15+28x-28<400,…28x<400+28-15,…x<413/28,…x<14.75
x=14

Originally posted by exc4libur on 12 Jul 2019, 08:29.
Last edited by exc4libur on 15 Jul 2019, 11:11, 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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12 Jul 2019, 08:36
Just simply check the options and conclude answer .
Only 1 option gives remainder of 3 on dividing by 4 and remainder 1 on diving by 7 i.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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12 Jul 2019, 08:45
Testing the answers, 15 is the only number in the option that will be divided by 4 to give a remainder of 3 and will also be divided by 7 to give a remainder of 1.

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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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12 Jul 2019, 08:52
IMO C

Given, n=4q1+3 [where q1 is quotient] => so values of n = 3,7,11,15,19,23,...so on
Also given, n=7q2+1 [where q2 is quotient] => so values of n = 1,8,15,22,29,36,...so on
The common lowest value is 15 and L.C.M of 4 and 7 is 28
Therefore we can calculate the values of n by the formula n=28k+15 (where k=0,1,2,3,4,...so on)
We can now check by the options, from (A) k=10, n=280+15=295 (Too small), [Usually we can try with the middle option C for back calculations)
Now from (C) k=13, n=28*13+15=379 (Close!)
Let's try one more, from (D) k=14, n=392+15=407 (Greater than 400)
C is our winner here.
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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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12 Jul 2019, 08:57
n=4k+3
n=7m+1
n<400

when n=15, the above conditions match.

Ans. (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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12 Jul 2019, 09:05
We can find the result easily by the options.From the first statement that when divided by 4 it leaves remainder 3 for sure its a odd no and therefore 3 options A,B,D goes out.
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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12 Jul 2019, 09:12
A. 10
B. 12
C. 13
D. 14
E. 15
Backsolving is best. Only 15/4 gives 3 as remainder. Therefore divide 15/7. Remainder is 1. So, 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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12 Jul 2019, 09:15
n<400
n= 4q+3.......(1)
n=7p+1........(2)

n=4q+3=7p+1
4q= 7p-2 —> q= (7p-2)/4

When p=2 , q= (7(2)-2)/4 =3
.: p=2 ,q=3

Substitute p=2 ,n=7(2)+1 =15
Again q=3 ,n=4(3)+3=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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12 Jul 2019, 09:18
The first such number is 15.

Add multiple of 4*7 in 15 to arrive at next such number. =43 and so on

Basically 15+28*13=379. 13 such values are possible.

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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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12 Jul 2019, 09:27
The general formula for n would be 28x+15. x can be any positive integer that makes 28x+15 <400. Solving this gives us 28x<385. The least integer value for x that will be < 385 is 13.
Re: If n is a positive integer less than 400, what is the number of n such   [#permalink] 12 Jul 2019, 09:27

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