This methodology does not always work. It worked out this time because you divided both numbers by a base 10 multiple of 2 and the greatest common prime factor was greater than 2.

However, consider 1010 (10 in base 10) and 100 (4 in base 10). The largest common prime factor in base 10 is 2.

If you simplify in binary in the same manner as you did for this question, you get 101 (5) and 10 (2). Now the numbers don't share any prime factors. You "took out" the 2 when you simplified.

Nice work william!

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Jamie Nelson
ManhattanGMAT Instructor

Not to dredge up this old thread, but where in the problem itself does it specify that 100010000 and 1000100000 are actually written in binary? It notes that the answers are written in binary, but not that the two large numbers above are -- if I simply said "Is this number a factor of 100010000?", I could be asking whether the number is a factor of 100,010,000, no? I'm not sure that the lack of commas is sufficient, especially given that you felt the need to clarify that the answer choices are in binary.

I get the intent of the problem -- we're supposed to take 100010000 as binary and translate it, etc. -- but I'm wondering if there's enough in the stem to warrant that assumption given that the explicit warning for the answer choices could be taken to set up a contrast to the last two numbers given. ("The answers are written in binary, but these aren't.") 17 isn't a factor of 100,010,000, so as written the answer could be (C), no, or is the lack of commas sufficient to state that a number must be written in binary?

Rather than say no, I would say not quite. That is, given two binary numbers with n matching trailing zeros in a situation like this, you can strip out the trailing zeros if, as indicated by William, you keep track of the additional factor for each number of 2^n (at least 'in your head'). That is, as binary numbers,
100010000 = 2^4 * 10001
and
1000100000 = 2^4 * 100010

Now you can look at the numbers the way you looked at them. Oh, and note the mixed base, the 2^4 is base 10 and the other numbers are binary. Of course you could write it as 10^100 in binary

Actually this works for any base. A number in base a with n trailing zeros following the number b can be written as a^n * b. An 'obvious' example is generally recognized by everyone if a is 10.

Is there a reason why the actual problem doesn’t show up in this thread?

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Tim Sanders
Manhattan GMAT Instructor