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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If n is a positive integer less than 400, what is the number of n such

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Intern  B
Joined: 24 Mar 2019
Posts: 22
If n is a positive integer less than 400, what is the number of n such  [#permalink]

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1
When you have since question there is a general formula which we can use.
We are given n=4X+3 and n=7Y+1
And the first of this number is 15.

We will get these number using this general equation

Number=(4*7)a+ 15
Number =28a+15

If we substitute the value of a from 1 we can find 13 more values before it passes the value of 400. So 13+1=14 is the answer

Posted from my mobile device

Originally posted by DarshBakshi on 12 Jul 2019, 08:48.
Last edited by DarshBakshi on 13 Jul 2019, 06:03, edited 1 time in total.
Manager  S
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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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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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n=4k+3
n=7m+1
n<400

when n=15, the above conditions match.

Ans. (E)
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Intern  B
Joined: 26 May 2018
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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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1

15/4=3._ and r=3; 15/7=2._ and r=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
Manager  G
Joined: 30 May 2019
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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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n=4q+3, so n can be 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43......................
n=7q+1, n can be 1, 8, 15, 22, 29, 36, 43..................
Common integers are 15, 43,....... and the difference is 43-15=28, this means that after 28 numbers we get one number that fits to both of our equations above. Accordingly, 15, 43, 71, 99, etc. We can manually add 28 and then we will finally reach some number less than 400 OR we can do a smart calculation by dividing 400/28 to see how many 28th we have in 400. It is a little more than 14, which is our answer (D)

Originally posted by mira93 on 12 Jul 2019, 09:02.
Last edited by mira93 on 13 Jul 2019, 02:09, edited 1 time in total.
Manager  G
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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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First number for "when n is divided by 4, the remainder is 3" is 3, second is 7, 11, 15, ,,,,,,,,,,,, 43 (add 4 to each term)

First number for "when n is divided by 7, the remainder is 1" is 1, second is 8, 15,........, 43 (add 7 to each term)

If we count manually we will observe pattern that we have same n for every other 14 numbers (or 28 in total). we can add 28 to 43 and see which numbers overlap. So, 43+28=71, 99, 127, 155, 183, 211, 239, 267, 295, 323, 351, 379. Overall 14 numbers.

While solving, however, I didn't manually added 28 to each number, I rather added to 43 (140, 28*5), so I got, 183, 323, 463 (nope, little more). Let's count till 323, we have 15, 43 2 numbers) 183 (since was multiplied by 5, we have 5 numbers in it), so 2+5+5=12 until 323. Add 56 to get 379 (our last number) and we have overall 12+2=14

Hence D
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Last edited by JonShukhrat on 13 Jul 2019, 05:28, edited 1 time in total.
Intern  B
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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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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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1
IMO-D

n=4p+3
n=7q+1

Therefore, 4p+3=7q+1 [p,q are positive integers ]
q=(4p+2)/7

p=3, q=2
p=10, q=6
similarly , p=17, 24, ......

n<400
4p+3<400
p max=99

p: 3,10,17,24,........99
3+(x-1)*7=99
x=96/7 +1 = 14 (nearest integer)
Manager  S
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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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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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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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Manager  B
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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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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?

Caution: We do NOT have to find A VALUE OF N. We have to find the NUMBER OF POSSIBLE VALUES of N that will satisfy the conditions. I made that mistake on my first attempt!

We need a value for N such that... N/7 has remainder 1, N/4 has remainder 3.

So it has to be 1 more than multiple of 7... say 8, 15, 22, 29 etc.
Also, it has to be 3 more than multiple of 4... say 7, 11, 15,19, 23, 27 etc...

The lowest value that satisfies this condition is 15 as it appears in both lists. We want to find all such values which will appear in both lists.

After listing few values I realised that values of N start with 15 and then increase by 28 (7x4).

So values are 15, 43, 71, 99, 127, 155, 183, 211, 239, 267, 295, 323, 351, and 379. (each value is previous value + 28.... and it stops at 379 because n is less than 400)

So there are 14 values of n which give remainder 3 when divided by 4 and 1 when divided by 7.

If there is a more elegant solution with some formula, I look forward to learning it tomorrow!
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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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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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1
1<=n<=400

Applying remainder theorem:
n-3 = 4k and n-1 = 7m (k,m are divisors)

n= 4K+3 and n=7m+1

Only Option E i.e. 15 fits ins.
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Manager  G
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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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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.
Manager  G
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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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n<400
N = 4Q1 + 3
N could be - 3, 7, 11, 15, 19

N = 7Q2 + 1
N could be 1, 8, 15, 22

So far we have found one number in the above list common 15 that satisfy both above equations and 15 is also in the answer choices

Manager  S
Joined: 09 Apr 2017
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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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Use Back-solved,

First, narrow down the answer choices using either N=4Q+3 or N = 7Q + 1

Pick, N = 7Q + 1 , since multiply of 7 is far less divisible compared to multiple of 4

A. 10 = 7Q+1, 9/7=Q not an integer
B. 12=7Q+1, 11/7=Q not an integer
C. 13=7Q+1, 12/7=Q not an integer
D. 14=7Q+1, 13/7=Q not an integer
E. 15=7Q+1, 14/7=Q an integer; definitely, 15=4Q+3, 12/4=Q is an integer

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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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 reminder = 4 ,then n=4K+3 = 3,7,11,15,19
n/7 reminder= 1 , n = 7k+1 = 1,8,15,22

n = 15 , Option E
Manager  S
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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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The first no. We encounter is 15

So next is 7*6
7*10+1
7*14+1
7*18+1
......
7*54+1

We have 15 such numbers (same can be done via 4)

E is correct

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Intern  B
Joined: 10 Nov 2018
Posts: 3
Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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There are two methods to solve this question.

First method :

Look at the answer choices and observe that only number 15gives the required remainders when divided by 4amd 7

The second way is to find the pattern of such numbers.. Luckily no 15 is first such number.. The next will be 43...

Posted from my mobile device
Intern  B
Joined: 10 Nov 2018
Posts: 3
Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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There are two methods to solve this question.

First method :

Look at the answer choices and observe that only number 15gives the required remainders when divided by 4amd 7

The second way is to find the pattern of such numbers.. Luckily no 15 is first such number.. The next will be 43...

Posted from my mobile device Re: If n is a positive integer less than 400, what is the number of n such   [#permalink] 12 Jul 2019, 10:05

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