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Re: If n is a positive integer less than 400, what is the number of n such
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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
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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(k1) 357/28=k1 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
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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
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13 Jul 2019, 02:18
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
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13 Jul 2019, 02:37
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
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13 Jul 2019, 03:12
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
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13 Jul 2019, 03:18
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
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Updated on: 13 Jul 2019, 05:21
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



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Re: If n is a positive integer less than 400, what is the number of n such
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13 Jul 2019, 04:04
\(n = 4p+3\) 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
\(96>3+(m1)*7\) \(m = 14\)



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Re: If n is a positive integer less than 400, what is the number of n such
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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. (39315)/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
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13 Jul 2019, 05:38
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)
ANSWER: D



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Re: If n is a positive integer less than 400, what is the number of n such
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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
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13 Jul 2019, 06:18
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
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13 Jul 2019, 06:31
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
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13 Jul 2019, 06:39
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 , Hence answer choice D must be the correct answer.
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If n is a positive integer less than 400, what is the number of n such
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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
Answer is D
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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
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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
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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
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13 Jul 2019, 07:37
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.
Answer choice is D



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Re: If n is a positive integer less than 400, what is the number of n such
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13 Jul 2019, 07:39
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
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