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# How many numbers lie between 11 and 1111 which when divided by 9 leave

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Joined: 05 Mar 2008
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How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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03 Nov 2009, 12:13
14
00:00

Difficulty:

65% (hard)

Question Stats:

62% (02:28) correct 38% (03:17) wrong based on 69 sessions

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How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

A.16
B.17
C.18
D.19
E.20

I looked for the first three common numbers and found the answer to be 33, 96, 159 (all separated by 63)

I set up the equation 63x + 33 starting with x = 0
Then I substituted the numbers above in the equation and found that x<18 because if x>=18 then the number is greater than 1111

so 0<= x <=17

Count the integers 17-0 + 1 = 18
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Re: How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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03 Nov 2009, 22:14
I solved it little differently. I took the first no. by multplying 21 by 1 and adding 12. i get the result 33 which fits the second test as well. then, 21 by 2 add 12 does not fit the test ans so do 21 by 3 + 12. However, 21 by 4 +12 does. that means the number has to be in the sequence with 1,4,7,10... 52 multiples of 21 as the number beyond 52 is outside the given set of numbers.

52+1/3 must be >17 hence the answer should be 18.
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Re: How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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04 Nov 2009, 00:28
2
1
Oh!

It is but natural that the numbers we are interested in are separated by 63 LCM of 9 and 21.

The first number is 33 and all others are separated by 63. It forms an arithmatic progression.

I am using the nth term formula of AP
nth term = a+ (n-1)d where a=33 and d=63.
the requirement is nth term less than 1111
hence a+(n-1)d<1111
33+(n-1)*63 < 1111
n-1<1078/63
n-1<17.11
n<18.11
the 17th term as per the formula a+(n-1)d is 1041
and the 18th term is 1104
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Re: How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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09 May 2011, 03:51
1
lcm = 63
a = 33

a+(n-1) d < 1111

giving n = 18
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Re: How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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09 May 2011, 04:59
2
N = 9K + 6 = 6,15,24,33,

N = 21m + 12 = 12, 33..

=> N = 63n + 33

Max value of N < 1111 = 63* 17 + 33

So total # = 18

Because n = 0 is the first number's index.

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Re: How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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25 Jan 2018, 04:36
lagomez wrote:
How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

A.16
B.17
C.18
D.19
E.20

I looked for the first three common numbers and found the answer to be 33, 96, 159 (all separated by 63)

I set up the equation 63x + 33 starting with x = 0
Then I substituted the numbers above in the equation and found that x<18 because if x>=18 then the number is greater than 1111

so 0<= x <=17

Count the integers 17-0 + 1 = 18

Check this link for more on how to solve such questions:
https://www.veritasprep.com/blog/2011/0 ... s-part-ii/
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Joined: 04 Mar 2011
Posts: 2815
Re: How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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29 Jan 2018, 10:38
lagomez wrote:
How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

A.16
B.17
C.18
D.19
E.20

The first number that leaves a remainder of 6 when divided by 9 is 6.

Then we have 6, 15, 24, 33, 42, 51, 60, and so on.

The first number that leaves a remainder of 12 when divided by 21 is 12.

Then we have 12, 33, 54 and so on.

We can see that 33 is the smallest number that satisfies both conditions.

Since the LCM of 9 and 21 is 63, we can keep adding 63 to 33 to find the succeeding numbers that satisfy both properties:

96, 159, ...

While we could list the numbers until we reach but not exceed 1111, a more efficient way is to find the find the largest number less than 1111 that satisfies both conditions, by the following equation:

33 + 63n < 1111

63n < 1078

n < 17.11

Since n is an integer, n = 17. That is, 33 + 63(17) is the largest integer that satisfies both conditions. Since 33 + 63(0) = 33 is the smallest integer that satisfies both conditions, we have 18 integers (in the form of 33 + 63m, where m is an integer from 0 to 17, inclusive) that satisfies both conditions.

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Re: How many numbers lie between 11 and 1111 which when divided by 9 leave  [#permalink]

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12 Feb 2019, 19:50
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Re: How many numbers lie between 11 and 1111 which when divided by 9 leave   [#permalink] 12 Feb 2019, 19:50
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