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# If the highest common factor of 2,472, 1,284 and positive integer N is

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Math Expert
Joined: 02 Sep 2009
Posts: 59721
If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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12 Nov 2019, 02:50
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Difficulty:

95% (hard)

Question Stats:

46% (02:30) correct 54% (02:26) wrong based on 78 sessions

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If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?

(A) $$2^2 * 3^2 * 7$$

(B) $$2^2 * 3^3 * 103$$

(C) $$2^2 * 3^2 * 5$$

(D) $$2^2 * 3 * 5$$

(E) None of these

Are You Up For the Challenge: 700 Level Questions

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Joined: 02 Aug 2009
Posts: 8310
Re: If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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12 Nov 2019, 07:38
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?

(A) $$2^2 * 3^2 * 7$$

(B) $$2^2 * 3^3 * 103$$

(C) $$2^2 * 3^2 * 5$$

(D) $$2^2 * 3 * 5$$

(E) None of these

Are You Up For the Challenge: 700 Level Questions

$$2472=2*2*2*3*103=2^3*3*103$$.
$$1284=2*2*3*107=2^2*3*107$$.

LCM=$$2^3*3^2*5*103*107$$
we can check each prime number
2 -- there can be two or 3 2s...$$2^2$$ or $$2^3$$
3 -- Surely two 3s as LCM has two 3s but none of 1284 or 2472 have two 3s...$$3^2$$
5 -- Surely one 5..$$5^1$$
103 -- can be none or one of 103...$$103^0$$ or $$103^1$$
107 -- can be none or one of 107...$$107^0$$ or $$107^1$$

As there are two 3s, A, B and D are out...
N can be any of -
$$2^2*3^2*5$$ and any of the combination of 2, 103 or 107 added to it..
for example $$2^3*3^2*5*103*107$$ can be the largest value

C

Note : The question should ask for the smallest value of N or it should be ' What can be the value of N?'
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If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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13 Nov 2019, 04:21

Solution

Given

• HCF of 2,472, 1,284 and positive integer N is 12.
• LCM of 2,472, 1,284 and positive integer N is 2^3∗3^2∗5∗103∗107.

To find

• The value of N.

Approach and Working out

• 2472 = 2^3 * 3 * 103
• 1284 = 2^2 * 3 * 107
• HCF (2472, 1284, N) = 12
o So, N = 2^(2+b) * 3^a * other prime factors where a >= 1 and b>=0

• LCM (2472, 1284, N) = 2^3∗3^2∗5∗103∗107.
o So, N can be 2^(2+b) * 3^2 * 5 * 103^c * 107^d where:
 b, c, d can be 0 or 1

Only possible option is option C.

Thus, option C is the correct answer.
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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15 Nov 2019, 08:01
EgmatQuantExpert wrote:

Solution

Given

• HCF of 2,472, 1,284 and positive integer N is 12.
• LCM of 2,472, 1,284 and positive integer N is 2^3∗3^2∗5∗103∗107.

To find

• The value of N.

Approach and Working out

• 2472 = 2^3 * 3 * 103
• 1284 = 2^2 * 3 * 107
• HCF (2472, 1284, N) = 12
o So, N = 2^(2+b) * 3^a * other prime factors where a >= 1 and b>=0

• LCM (2472, 1284, N) = 2^3∗3^2∗5∗103∗107.
o So, N can be 2^(2+b) * 3^2 * 5 * 103^c * 107^d where:
 b, c, d can be 0 or 1

Only possible option is option C.

Thus, option C is the correct answer.

HCM X LCM = Product of numbers

2472 X 1284 X N = 12 X 2^3 X 3^2 X 5 X 103 X 107

solving it results in N = 3*5 = 15
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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20 Nov 2019, 18:58
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?

(A) $$2^2 * 3^2 * 7$$

(B) $$2^2 * 3^3 * 103$$

(C) $$2^2 * 3^2 * 5$$

(D) $$2^2 * 3 * 5$$

(E) None of these

Are You Up For the Challenge: 700 Level Questions

Let’s first factor 2,472 and 1,284 into primes:

2,472 = 12 x 206 = 2^2 x 3 x 2 x 103 = 2^3 x 3 x 103

1,284 = 12 x 107 = 2^2 x 3 x 107

We see that that the LCM of 2,472 and 1,284 is 2^3 x 3 x 103 x 107. Notice that the LCM of 2,472 1,284, and N is equal to the LCM of 2^3 x 3 x 103 x 107 and N, which is given to be 2^3 x 3^2 x 5 x 103 x 107. We are also given that the GCF of 2,472 1,284, and N is 12.

Recall that for any two positive integers a and b, we have a x b = LCM(a, b) x GCF(a, b). Therefore, by letting a = 2^3 x 3 x 103 x 107 and b = N, we have:

2^3 x 3 x 103 x 107 x N = 2^3 x 3^2 x 5 x 103 x 107 x 12

N = 3 x 5 x 12

N = 2^2 x 3^2 x 5

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# Scott Woodbury-Stewart

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Re: If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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01 Dec 2019, 06:15
chetan2u wrote:
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?

(A) $$2^2 * 3^2 * 7$$

(B) $$2^2 * 3^3 * 103$$

(C) $$2^2 * 3^2 * 5$$

(D) $$2^2 * 3 * 5$$

(E) None of these

Are You Up For the Challenge: 700 Level Questions

$$2472=2*2*2*3*103=2^3*3*103$$.
$$1284=2*2*3*107=2^2*3*107$$.

LCM=$$2^3*3^2*5*103*107$$
we can check each prime number
2 -- there can be two or 3 2s...$$2^2$$ or $$2^3$$
3 -- Surely two 3s as LCM has two 3s but none of 1284 or 2472 have two 3s...$$3^2$$
5 -- Surely one 5..$$5^1$$
103 -- can be none or one of 103...$$103^0$$ or $$103^1$$
107 -- can be none or one of 107...$$107^0$$ or $$107^1$$

As there are two 3s, A, B and D are out...
N can be any of -
$$2^2*3^2*5$$ and any of the combination of 2, 103 or 107 added to it..
for example $$2^3*3^2*5*103*107$$ can be the largest value

C

Note : The question should ask for the smallest value of N or it should be ' What can be the value of N?'

Hi, Can you plz explain why cant we follow the method:

HCF * LCM = Product of numbers?

If we use that, it results in N=15
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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01 Dec 2019, 07:46
ScottTargetTestPrep wrote:
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?

(A) $$2^2 * 3^2 * 7$$

(B) $$2^2 * 3^3 * 103$$

(C) $$2^2 * 3^2 * 5$$

(D) $$2^2 * 3 * 5$$

(E) None of these

Are You Up For the Challenge: 700 Level Questions

Recall that for any two positive integers a and b, we have a x b = LCM(a, b) x GCF(a, b). Therefore, by letting

a = 2^3 x 3 x 103 x 107 and b = N, we have:

2^3 x 3 x 103 x 107 x N = 2^3 x 3^2 x 5 x 103 x 107 x 12

N = 3 x 5 x 12

N = 2^2 x 3^2 x 5

can you explain the bold line?

2472 x 1284 x N = (2^3 x 3 x 103) x (2^2 x 3 x 107) x N

arnt you missing a few terms?
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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03 Dec 2019, 20:15
Mansoor50 wrote:

can you explain the bold line?

2472 x 1284 x N = (2^3 x 3 x 103) x (2^2 x 3 x 107) x N

arnt you missing a few terms?

I did not intend to multiply all three numbers together; as a matter of fact, when you have more than two numbers, the formula a x b = LCM(a, b) x GCF(a, b) is no longer valid.

We are taking a = LCM(2472, 1284) and b = N and applying the formula to these numbers.
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

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Joined: 29 May 2017
Posts: 142
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: If the highest common factor of 2,472, 1,284 and positive integer N is  [#permalink]

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05 Dec 2019, 07:17
ScottTargetTestPrep wrote:
Mansoor50 wrote:

can you explain the bold line?

2472 x 1284 x N = (2^3 x 3 x 103) x (2^2 x 3 x 107) x N

arnt you missing a few terms?

I did not intend to multiply all three numbers together; as a matter of fact, when you have more than two numbers, the formula a x b = LCM(a, b) x GCF(a, b) is no longer valid.

We are taking a = LCM(2472, 1284) and b = N and applying the formula to these numbers.

WOOT!......THANKS!!!!!
Re: If the highest common factor of 2,472, 1,284 and positive integer N is   [#permalink] 05 Dec 2019, 07:17
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