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

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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, 08:00
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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


 

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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: 12 Jul 2019, 20:02
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n is divided by 4, the remainder is 3 and when n is divided by 7, the remainder is 1

The numbers would be 15, 43, 71,.....
Its an AP with difference as 28. -> 15 + (n-1)*28

Among the options, if there are 13 numbers, the 13th number would be
15 + 12*28 = 351.

The 14th number would be 15 + 13*28 = 379.

15th no = 407

14 numbers less than 400.

Option D

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Originally posted by prashanths on 12 Jul 2019, 08:21.
Last edited by prashanths on 12 Jul 2019, 20:02, 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, 08:28
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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=4k+3; 3,7,11,15,...
n=7m+1; 1,8,15,22...

Then
n=28t+15; 15....
400-15=385
385/28 = 13,...
Don't forget about 15.
13+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, 11:17
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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
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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

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, 08:12
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it follows the pattern of 2,,10.....54th multiples of 7......till 57*7=399...... total is 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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New post 12 Jul 2019, 08:13
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IMO Answer is D

First number which satisfies the condition is 15, second number which satisfies the condition is 43,

sequence is 15,43,-----

15+(n-1)*28 < 400

n which satisfies the equation is 14.

SO, 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, 08:21
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n=4x+3 and n=7y+1

4x+3=7y+1

y=4x+2/7

We need to check for how many values of x we get an integer for y. Also note that x must be less than 100 since if x=100, n>400

Going by patterns, we get an integer solution for y for every 7th value of x from x=3 which is a total of 14

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, 08:27
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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

Numbers such that when n is divided by 4, the remainder is 3; 3,7,11,15,19,23,27...
Numbers such that when n is divided by 7, the remainder is 1; 1,8,15,22,29,36,43....

First n = 15
Next n = 15 + 7*4 = 15 +28 = 43
It is an arithmetic progression with a = 15 and d=28
Last such number < 400 = 15 + 13*28 = 379
n=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, 08:30
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Remainder of 1 when divided by 7 - 1,8,15,22,29,36,43,50,57,64,71....
Of the above 15,43,71 leaves a remainder of 3 (the previous number to the multiple of 7 must have only one multiple of 2, 15-7*2+1, 43-7*2*3+1, 71-7*5*3+1).

Number of multiples of 7 in 400 = 400/7=57 and 54 is the last multiple below 400 of the form 7*27*2+1.

Tn=54 ; 54=2+(n-1)4 using AP. N=14. so 14 terms will satisfy the requirement. 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, 08:33
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given
A= n*4+3
and A= n*7+1
we see that A= 15,43,71 ... so on
d= 28 ,a=15
an=400
400=15+(n-1)*38
n=14.75 for n = 413 but n<400 so n = 14
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
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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, 08:38
1
Answer is D

According to the given condition the numbers are: 15,43,71,99,127...We see a nice pattern here and that is the difference between the numbers is 28 so we do not need to calculate the numbers till 400. Since N<400 so the numbers that satisfies the conditions will be 14.
[Numbers are: 15,43,71,99,127,155,183,211,239,267,295,323,351,379]

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, 08:41
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n<400
n = 4k+3 = 3,7,11,15,.....43,.......71,....99

n = 7k+1 = 1,8,15,.....43.....,71,.....99

number common in both 15,43,71,99

we get an AP here first tern 15 and common difference = 28
total number of such numbers <400

let a be the first term and d= common difference b=total number of such term
then bth term can be found by =a+(b-1)d (and this term should be less than 400)

a+(b-1)d<400
15+(b-1)28<400
(b-1)28<385
(b-1)<13.75
b<14.75
b= 14

so the total number of such integer = 14
D is the 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, 08:42
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LCM of 7 and 4 is 28

The first number n can be is 15

Now we need to check how many times can we add 28 to 15 and be under 400

Answer is D) 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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New post 12 Jul 2019, 08:43
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n = 4A+3
n= 7B+1

By quick hit and trial, the first number to satisfy the condition is "15"
The next number in the progression would be 15+ LCM(4,7)==> 28

It will turn out be an AP series

where a = 15 d=28 and the max value<400

15, 43,71....... maxvalue<400

solving this will give the answer as 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, 06:03
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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

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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.
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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, 08:59
1
Answer is D

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
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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, 02:09
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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.
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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, 05:28
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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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Originally posted by JonShukhrat on 12 Jul 2019, 09:04.
Last edited by JonShukhrat on 13 Jul 2019, 05:28, 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, 09:08
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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)
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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, 09:08

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