Every digit of a number written in binary is either 0 or 1. To transla

MANHATTAN GMAT OFFICIAL SOLUTION:

To begin this problem, translate the numbers written in binary.

\(100010000 = 1*2^8 + 0*2^7 + 0*2^6 + 0*2^5 + 1*2^4 + 0*2^3 + 0*2^2 + 0*2^1 + 0*2^0\)

This simplifies to \(2^8 + 2^4\).

Similarly, 1000100000 simplifies to \(2^9 + 2^5\).

We could then translate these further.

\(2^8 + 2^4 = 256 + 16 = 272\)
\(2^9 + 2^5 = 512 + 32 = 544\)

We need to find the greatest common factor of both of these numbers. At this point, we could do a prime factorization of both numbers. They are large, however, and this process could be tedious. It will actually be easier to find common factors of these numbers from their previous forms.
When an expression like 2^8 + 2^4 appears on the GMAT, it is usually helpful to factor out the largest common factor of both terms of the expression. In this case, factor out 2^4.

\(2^8 + 2^4\) → \(2^4(2^4+1)\) → \(2^4(16+1)\) → \(2^4(17)\)

From this we can see that 272 has only two prime factors: 2 and 17.

Now do the same for 544.

\(2^9 + 2^5\) → \(2^5(2^4 + 1)\) → \(2^5(16 + 1)\) → \(2^5(17)\)

544 has the same two prime factors: 2 and 17. So the greatest common prime factor of the two numbers is 17. Now we need to translate 17 back into binary. Notice from the manipulations above that

\(17 = 2^4 + 1, or 2^4 + 2^0\)

Thus, 17 written in binary is 10001.

The correct answer is E.

Binary Divison can provide a quick answer if you are comfortable with it.

as option E is the biggest binary number we try with it first :

100010000/ 10001 =10000

1000100000/ 10001 =100000

so answer is option is E

Thanks for the explanation Bunuel. But do we really need to go through this process. I just looked at the two numbers in the question. The second number ( 1000100000) is the first number (100010000) multiplied by 10. Then I took a look at E which is the highest value given among the questions. I just realized that you would need to multiply that by 10 several times to divide both the first number and the second number so I chose E. I didn't bother with the binary calculations. Is my reasoning flawed?

The question stem is weirdly worded.


Okay, and then what? You should clarify that you should then sum those results to get the number in base 10.

Note that you are missing out on an important aspect in this method. You need the largest prime number. You need to verify that 10001 is prime.

Seems that this post is already well discussed, but let me given by approach

Firstly Binary is beautiful concept, Here Indexing plays a important role while extracting or converting numbers from binary to decimal

Let me index it first

1. 100010000

10 9 8 7 6 5 4 3 2 1 0
-0 0 1 0 0 0 1 0 0 0 0

That index is raised to power of 2 while computing Decimal = 2^8*1 + 2^0*1 = 256+16 = 272


2. 1000100000

10 9 8 7 6 5 4 3 2 1 0
-0 1 0 0 0 1 0 0 0 0 0

That index is raised to power of 2 while computing Decimal = 2^9*1 + 2^1*1 = 512 +32 = 544


As numbers are multiple of each other with a factor two 272*2 = 544

So the factors of 272 = {2,2,2,2,17}
largest prime is 17



Now converting decimal into binary
Index is raised to power two then we get the numbers
- 5 4 3 2 1 0 == index
- 32 16 8 4 2 1 == Raise to power the index
- 0 1 0 0 0 1 == 17


So answer is E 10001

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One more thing numbers 100010000 & 1000100000 are in relation with each other as in binary when a digit is moved right ways (as 1 changed its index from 8 to 9 & from 4 to 5) it gets multiplied by 2 in decimal & when a digit is moved left ways it gets divided by 2 in decimal

Hi Bunuel
I believe the question is incomplete. After mentioning multiplication by 2^(n-1), it needs to mention that 0/1 *2^(n-1) needs to be added for each digit. If I do not know / remember binary to integer conversion, I wont be able to solve this question.

Thanks,
­

Question does not describe, at all, that digits should be added. For all I can understand, the first number is 256000160000. Where exactly am I told that 256 and 16 should be added?