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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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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

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

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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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Updated on: 12 Jul 2019, 20:02
6
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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12 Jul 2019, 08:09
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

15 is the number which leaves 3 and 1 with 4 and 7 respectively.

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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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12 Jul 2019, 08:11
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?

given:
n = 4*p + 3=7*q + 1, where p & q are integers

A. 10 --> 10 = 4*2 + 2
B. 12 --> 12 = 4*3 + 0
C. 13 --> 13 = 4*3+1
D. 14 --> 14 = 4*3+2
E. 15 --> correct: 15 = 4*3 + 3 = 7*2+1
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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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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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12 Jul 2019, 08:13
1
1

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.

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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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12 Jul 2019, 08:19
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

The values that satisfy both the condition are 15,43,71.
So if we write the equation the number has to satisfy 4x+3 and 7y+1. The values of y which satisfy both the equation are y=2,y=6,y=10,14,18 so we see that it jumps by 4. so last number is going to be y=42 which when put into equation number turns out to be 54.
hence total numbers to satisfy the condition is 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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12 Jul 2019, 08:21
1
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

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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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12 Jul 2019, 08:27
1
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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12 Jul 2019, 08:28
C

1st condition: n = 4a+3 = 3,7,11,15,19,23...
2nd condition: n=7b+ 1 = 1,8,15,22,...

So, first common integer in both series is 15. GCD of 4 and 7 is 28.

Therefore, n = 28*x + 15. Now, since n has to be less than 400, we have: 28*x + 15 < 4000 => x < 13.5..
So maximum value of x is 13.
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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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12 Jul 2019, 08:28
3
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,...
13+1=14

IMO D
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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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Updated on: 15 Jul 2019, 11:11
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

0 < n positive < 400
when n/4 remainder is 3: n={3 7 11 [15] 19 23…[43]…}
when n/7 remainder is 1: n={1 8 [15] 22 29 36…[43]…}
the pattern of n such numbers that fit the condition has a common difference of 28 [=43-15]
thus we need to find the last term with a common difference of 28 from 0 to 400,
with a first term of 15 and a last term of 15+28(x-1)<400:
15+28(x-1)<400,…15+28x-28<400,…28x<400+28-15,…x<413/28,…x<14.75
x=14

Originally posted by exc4libur on 12 Jul 2019, 08:29.
Last edited by exc4libur on 15 Jul 2019, 11:11, 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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12 Jul 2019, 08:30
1
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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12 Jul 2019, 08:33
1
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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12 Jul 2019, 08:36
Just simply check the options and conclude answer .
Only 1 option gives remainder of 3 on dividing by 4 and remainder 1 on diving by 7 i.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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12 Jul 2019, 08:38
1

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]

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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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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
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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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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

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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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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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Re: If n is a positive integer less than 400, what is the number of n such  [#permalink]

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12 Jul 2019, 08:45
Testing the answers, 15 is the only number in the option that will be divided by 4 to give a remainder of 3 and will also be divided by 7 to give a remainder of 1.

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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, 08:45

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