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Which of the following represents the largest 4 digit number

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Which of the following represents the largest 4 digit number  [#permalink]

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New post 06 Jun 2011, 03:14
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Which of the following represents the largest 4 digit number that can be added to 7249 in order to make the derived number divisible by each of 12, 14, 21, 33 & 54.

A. 9123
B. 9383
C. 8727
D. 1067
E. None Of The Above
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Re: LCM & HCF  [#permalink]

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New post 06 Jun 2011, 03:27
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mattapraveen wrote:
Which of the following represents the largest 4 digit number that can be added to 7249 in order to make the derived number divisible by each of 12, 14,21, 33 & 54.

A.9123
B.9383
C.8727
D.1067
E.None Of The Above


Find the LCM of the given numbers.
12= 2*2*3
14= 2*7
21= 3*7
33= 3*11
54= 3*2*3*3

Max powers of all prime numbers:
2=2
3=3
7=1
11=1

LCM=2^2*3^3*7^1*11^1=8316

8316-7249=1067. Thus, if we add 1067 to 7249, the number will be divisible by all the given numbers.
However, 1067 is not the GREATEST 4-digit number to satisfy the condition.

Next number that will be divisible is:
1067+8316=9383

Thus, adding 9383 to 7249 will give us a number that will be divisible by all the given numbers PLUS 9383 is the greatest 4-digit number that satisfies this condition.

Ans: "B"
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Re: LCM & HCF  [#permalink]

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New post 06 Jun 2011, 04:24
here,
LCM of 12,14,21.33 and 54 comes to be 8316.
Now subtract 7249 from 8316 we get 1067 but 1067 is n0t the greatest 4 digit number so add 1067 to 8316 we get 9383. thus "B" is the correct option.
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Re: LCM & HCF  [#permalink]

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New post 06 Jun 2011, 04:24
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fluke wrote:
mattapraveen wrote:
Which of the following represents the largest 4 digit number that can be added to 7249 in order to make the derived number divisible by each of 12, 14,21, 33 & 54.

A.9123
B.9383
C.8727
D.1067
E.None Of The Above


Find the LCM of the given numbers.
12= 2*2*3
14= 2*7
21= 3*7
33= 3*11
54= 3*2*3*3

Max powers of all prime numbers:
2=2
3=3
7=1
11=1

LCM=2^2*3^3*7^1*11^1=8316

8316-7249=1067. Thus, if we add 1067 to 7249, the number will be divisible by all the given numbers.
However, 1067 is not the GREATEST 4-digit number to satisfy the condition.

Next number that will be divisible is:
1067+8316=9383

Thus, adding 9383 to 7249 will give us a number that will be divisible by all the given numbers PLUS 9383 is the greatest 4-digit number that satisfies this condition.

Ans: "B"


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Re: LCM & HCF  [#permalink]

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New post 06 Jun 2011, 05:05
Let us do the prime factorization of the numbers and get the LCM

12 = 2^ * 3
14 = 2 * 7
21 = 7*3
33 = 3 * 11
54 = 3^3 * 2

So LCM = 3^3 * 11 * 7 * 2^2

= 27 * 77 * 4

= 8316

Now the lowest number to satisfy thus would be = 8316 - 7249 = 1067

But we need to check other options also, so let us add 8316 once again to 1067.

= 8316 + 1067 = 9383

Answer - B
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Re: LCM & HCF  [#permalink]

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New post 11 Jun 2011, 15:26
good one...nice explanation
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Re: LCM & HCF  [#permalink]

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New post 12 Nov 2013, 12:53
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mattapraveen wrote:
Which of the following represents the largest 4 digit number that can be added to 7249 in order to make the derived number divisible by each of 12, 14,21, 33 & 54.

A.9123
B.9383
C.8727
D.1067
E.None Of The Above


Trying to find a simple approach for this one.
Let's see if this works

So we need it to be divisible by 12, 14, 21, 33 and 54. Note that these numbers have several primes in common. What is the largest prime that we have among all these. 11.

OK so we need to find out if the sum will be divisible by 11. First we can tell that 7249 is divisible by 11 because 7+4-2-9=0 which is a multiple of 11.
So we need another multiple of 11, so that the sum of both also throws a multiple of 11. Let's strat with the greatest number cause that's what wer are being asked.
9383 = 17-6 = 11. It is of course a multiple of 11 so option (B) then is the correct answer

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Re: LCM & HCF   [#permalink] 12 Nov 2013, 12:53
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