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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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New post 13 Jul 2019, 07:53
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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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New post 13 Jul 2019, 07:54
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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



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

So the answer is 14

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 13 Jul 2019, 07:56
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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.

Answer (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 13 Jul 2019, 13:25
The number leaves a remainder of 3 when divided by 4.

So the number N can be written as

N = 4k+3, where integer k=0,1,2,3....

Also, N leaves a remainder of 1 on dividing by 7.

Therefore, {4k+3-1} is divisible by 7

or

4k+2 = 7m {where m is an integer}
=> m = (4k+2)/7

Applying remainder theorem we see that

rem(4k/7) = 5 since rem(2/7) = 2 and m is an integer

if we substitue k = 3+7h , we see that rem([12+21h]/7) = 5

hence we can write k as 3+7h

So the number N can be written as

N = 4(3+7h) + 3
=> N = 28h + 15

No. of multiples of 28 less than 400 = [400/28] = 14 {Where [] denotes the greatest integer function}

So h can take values from 0 to 13 {Since for h=14, N>400}

Hence total number of possible values of N becomes 0 to 13 , i.e. 14

So 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 14 Jul 2019, 03:04
Bunuel wrote:
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


 

This question was provided by Math Revolution
for the Game of Timers Competition

 



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
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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 15 Jul 2019, 07:12
This is a good one. Basically you have to see that we are asked to find the number of N not a possible value for N!

The first possible value of n is 15. Once you have this you should see the sequence increases by 28 (4*7). Because n is < 400. You can look at how many times 28 goes into 400, you can guesstimate: 28*10 + 28*3 so 13 times plus the first number in the sequence (15) gives you a total of 14 terms. Hence 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] 15 Jul 2019, 07:12

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