Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
To begin this problem, translate the numbers written in binary.
100010000 = 1×28 + 0×27 + 0×26 + 0×25 + 1×24 + 0×23 + 0×22 + 0×21 + 0×20
This simplifies to 28 + 24.
Similarly, 1000100000 simplifies to 29 + 25.
We could then translate these further.
28 + 24 = 256 + 16 = 272
29 + 25 = 512 + 32 = 544
29 + 25 = 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 28 + 24 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 24.
28 + 24 ? 24(24 + 1) ? 24(16 + 1) ? 24(17)
From this we can see that 272 has only two prime factors: 2 and 17. Now do the same for 544.
29 + 25 ? 25(24 + 1) ? 25(16 + 1) ? 25(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 = 24 + 1, or 24 + 20. Thus, 17 written in binary is 10001.
The correct answer is E.
