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Manager  Joined: 07 Sep 2010
Posts: 245
The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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16 00:00

Difficulty:   95% (hard)

Question Stats: 45% (02:51) correct 55% (02:37) wrong based on 148 sessions

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The Least Common multiple of 2^6-1 and 2^9-1 is:

A. $$2^{12}+27*2^9-217$$

B. $$2^{12} +63*2^3-1$$

C. $$2^{12}+5^{29}-1$$

D. $$2^{12}+9*2^8 -1$$

E. None of these.

Originally posted by imhimanshu on 21 May 2013, 07:23.
Last edited by Bunuel on 08 Jul 2014, 01:35, edited 3 times in total.
Renamed the topic and edited the question.
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Re: The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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imhimanshu wrote:
Question

The Least Common multiple of $$2^6-1$$ and $$2^9-1$$ is

a) 2^12+27*2^9-217
b) 2^ 12 +63*2^3-1
c) 2^12+5^29-1
d) 2^12+9*2^8 -1
e) None of these.

Hi Experts,
I would like to know what is the best approach to solve such questions.

Help will be appreciated.

Thanks
H

Responding to a pm:

The best approach in my opinion is what Zarrolou has suggested above.

LCM * GCF = Product of the numbers = $$(2^6-1)*(2^9-1) = (2^3 - 1)(2^3 + 1) * (2^3 - 1)(2^6 + 1 + 2^3)$$

Notice that the only common factor between them is $$(2^3 - 1)$$ so this must be the GCF. Hence LCM will be the rest of the product.

$$LCM = (2^3 + 1) * (2^3 - 1)(2^6 + 1 + 2^3) = (2^6 - 1)(2^6 + 1 + 2^3)$$

Now how do you get it in the format in the options? Almost all options have $$2^{12}$$ and 1 so retain those two and club everything else together.

$$LCM = 2^{12} - 1 + (2^6 + 2^9 - 2^6 - 2^3) = 2^{12} - 1 + 2^3(2^6 - 1) = 2^{12} - 1 + 2^3*63$$

Note that option 'none of these' makes it more complicated since you cannot try some more esoteric methods e.g. last digit etc. GMAT doesn't give you this option. Also, GMAT doesn't expect you to know a^3 - b^3 = (a-b)(a^2 + ab + b^2). Of course, you should be able to arrive at the LHS, given the RHS.
Hence, there are very few CAT questions which will actually be GMAT relevant. If you are practicing for GMAT, try to stick to a GMAT source, especially if you have limited time.
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Re: The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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1
2
Factoralization (refer herefor the formulas)

$$2^6-1=(2^3+1)(2^3-1)=9*7=3*3*7$$

$$2^9-1=(2^3-1)(2^6+2^3+1)=7*73$$

$$LCM=9*7*73=(2^3+1)(2^3-1)(2^6+2^3+1)=(2^6-1)(2^6+2^3+1)=2^{12}+2^3(2^6-1)-1$$

b) $$2^ {12} +63*2^3-1$$
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Re: The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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imhimanshu wrote:
Question

The Least Common multiple of $$2^6-1$$ and $$2^9-1$$ is

a) 2^12+27*2^9-217
b) 2^ 12 +63*2^3-1
c) 2^12+5^29-1
d) 2^12+9*2^8 -1
e) None of these.

Hi Experts,
I would like to know what is the best approach to solve such questions.

Help will be appreciated.

Thanks
H

I feel [B] is the answer. I am not sure my way is the shortest (coz i did have to go thru a lot of calculation), but it is a way nonetheless From the above, one can find that the HCF of the numbers $$2^6-1$$ and $$2^9-1$$ is 7. By the rule,
LCM*HCF = Product of two numbers
=> LCM = Product of two numbers /7

7 can be expressed as $$2^3 -1$$. By checking all the answer choices one can see that only [B] i.e. $$2^ 12 +63*2^3-1$$,
also written as, $$2^ 12 +(2^6 - 1)2^3-1 = 2^ 12 +2^9 -2^3 -1$$, results the product of numbers i.e. $$2^15 - 2^6 - 2^9 + 1.$$

Hope my answer is accurate! Would certainly appreciate a shorter way on this! Regards,
Arpan

**edited to correct typo
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Re: The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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Bunuel, kindly update the OA in expanded / Mathematical form.
The un-formatted options tend to confuse Thanks
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Joined: 02 Sep 2009
Posts: 64951
Re: The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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PareshGmat wrote:
Bunuel, kindly update the OA in expanded / Mathematical form.
The un-formatted options tend to confuse Thanks

______________
Done.
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Re: The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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The Least Common multiple of 2^6-1 and 2^9-1 is:

The following steps will help to solve the sum very easily-:

First, lets get the units digit values of 2^6-1 and 2^9-1
1.Units digit of 2^6-1 is 3
2.Units digit of 2^9-1 is 1

Consider every option here with respect to taking the units digit of the value.
A. - 5 (Eliminated as the required values are not multiples of 3 and 1)
B. - 3
C. - 0 (Eliminated as the required values are odd numbers, i.e. 3 and 1)
D.- 9

We are left with option B and D

LCM of 3 and 1 is option B., i.e. 3( which is its unit digit)

Thanks.

Regards,
Raunak Damle!

Cheers!
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The Least Common multiple of 2^6-1 and 2^9-1 is  [#permalink]

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imhimanshu wrote:
The Least Common multiple of 2^6-1 and 2^9-1 is:

A. $$2^{12}+27*2^9-217$$

B. $$2^{12} +63*2^3-1$$

C. $$2^{12}+5^{29}-1$$

D. $$2^{12}+9*2^8 -1$$

E. None of these.

The Least Common multiple of 2^6-1 and 2^9-1 is:

$$2^6 -1 = (2^3+1)(2^3-1)$$
$$2^9 -1 = (2^3 -1)(2^6 + 2^3 + 1)$$

$$LCM (2^6 -1,2^9 -1) = (2^3+1)(2^9-1) = 2^{12} +2^9 -2^3 -1 = 2^{12} + 64*2^3 - 2^3 - 1 = 2^{12} +63*2^3 - 1$$

IMO B
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Email: kinshook.chaturvedi@gmail.com The Least Common multiple of 2^6-1 and 2^9-1 is   [#permalink] 30 Mar 2020, 09:38

# The Least Common multiple of 2^6-1 and 2^9-1 is  