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

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

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

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.

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)

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

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)

n = 4a+3 = {3, 7, 11, 15, 19, 23, 27, 31, ...}
also
n = 7b+1 = {1, 8, 15, 22, 29, 36, ...}

First common term = 15

Every terms after the first term will be at a gap of LCM (4 & 7) i.e. 28

Terms are {15, 43, 71....}

Last term = 15 + (n-1)*28 < 400

i.e. n-1 < 13.75
i.e. n < 14.75

i.e. Maximum value of n = 14

Answer: Option D

Used online whiteboard for this question and took 3 mins. Uff!

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