Shane04 wrote:
KarishaB wrote:
Sure you can take a number of the form and check (as you did) but the issue here is that it has an option 'None of these.' So what works for one example, may not work for all such numbers and hence, you need to establish it conceptually.
e.g. ababab = ab * 10101
Since 10101 is divisible by 3, no matter what ab is, ababab will be divisible by 3.
That said, GMAT does not give this option "None of these" so normally you don't have to worry about it.
Thank you for your response.
I get the conclusion that 3 can divide 10101, but can you explain how do we arrive at 10101*ab?
Also if I chose 101010 as the minimum value instead of 111111, and reduced it to its primes, would it then be correct to determine that 3 will always be a divisor and hence, a factor.
Eg: 101010=2*3*5*7*13*37
You could take any example - it needn't be the smallest such number. (As an aside, note that in this case, since a and b are positive integers, 101010 is not the smallest such number since b cannot be 0.)
A number which must be a factor of all numbers of the form ababab will be a factor of your example too. Hence you can pick any number of the form ababab. Note that it is not necessary that if the smallest such number has that factor then every greater number will also have that factor. For example, 11 is a factor of 111111 but it will not be a factor of 636363.
The point is that if we are looking for a factor of ababab such that this factor is a factor of all such numbers, then it will be a factor of 111111, 131313, 878787, 929292 etc. So I could pick any number, find its factors and then try to match it with the given options.
If I had picked 929292, then I would get 3, 4 and 6 as factors. So I would need to pick another number such as 878787 to confirm that 4 and 6 are not necessarily factors of all numbers of the form ababab.
As for how to figure out that ababab = ab * 10101, it is all about pattern recognition.
We see that aaa = a * 111 (try the multiplication)
Then if we have 2 digits getting repeated, ababab = ab * 10101 (again, try the multiplication by taking values for a and b)
Similarly, abcabc = abc * 1001
I suggest you to check out these posts and videos on factors and factorisation:
Factors:
https://anaprep.com/number-properties-f ... -a-number/https://anaprep.com/number-properties-r ... e-factors/https://youtu.be/DxIH8rjhpKYFactorisation
https://youtu.be/Kd-4cH4cqHw