sanju09 wrote:Shaina's five-distinct-digit locker code is 5A48B. What digit letter A symbolizes in Shaina's locker code?
I. Shaina's locker code is divisible by all integers 2 through 6.
II. Shaina's locker code is divisible by 9 and 11.



[spoiler]made up by Sanjeev K Saxena for Avenues Abroad[/spoiler]

With statement 1, we know that Shaina's locker code is even and divisible by 5, so B must be 0. And since the code is divisible by 3, the sum of the digits must be divisible by 3.

Okay. So, the sum of the known digits is:

5 + 4 + 8 + 0 = 17

So, we will need to add 1, 4 or 7 to get a multiple of 3. (Anything else is a non-digit.) A could thus be 1, 4 or 7. Statement 1 is not sufficient.

With statement 2, the sum of the digits has to be a multiple of 9. The sum of the known digits is 5 + 4 + 8 = 17, so we'd need to add either 1 or 10 (anything else would involve adding a non-digit for either A or B) to get a multiple of 9.

We are left with a rather limited range of possible locker codes:

51480
50481

51488
52487
53486
54485
55484
56483
57482
58481

Which one(s) of these is a multiple of 11? The way we check is to add up the odd numbered digits and add up the even numbered digits, then subtract the first by the second and see whether we get a multiple of 11.

51480: 5 + 4 + 0 - (1 + 8) = 9 - 9 = 0 yes
50481: 5 + 4 + 1 - (0 + 8) = 10 - 8 = 2 no

51488: 5 + 4 + 8 - (1 + 8) = 17 - 9 = 8 no
52487: 5 + 4 + 7 - (2 + 8) = 16 - 10 = 6 no
53486: 5 + 4 + 6 - (3 + 8) = 15 - 11 = 4 no
54485: 5 + 4 + 5 - (4 + 8) = 14 - 12 = 2 no
55484: 5 + 4 + 4 - (5 + 8) = 13 - 13 = 0 yes
56483: 5 + 4 + 3 - (6 + 8) = 12 - 14 = 2 no
57482: 5 + 4 + 2 - (7 + 8) = 11 - 15 = 4 no
58481: 5 + 4 + 1 - (8 + 8) = 10 - 16 = 6 no

Our combination could thus be 51480 or 55484. Statement 2 is not sufficient.

Statements 1 and 2 combined only give us the possibility of 51480. Thus A = 1, and the two statements combined are sufficient. The correct answer would be C.

On a side note, be extremely careful about problems that you get from somewhere other than the official guide. I have never seen a real GMAT question that played off of the algorithm for multiples of 11 for numbers larger than 250 unless you could tell plainly from the digits (i.e., 1,100,110) as it takes FAR, FAR too long to realistically expect even a very bright GMAT student to get done in a reasonable amount of time.

rustypolymath wrote:

sanju09 wrote:Shaina's five-distinct-digit locker code is 5A48B. What digit letter A symbolizes in Shaina's locker code?
I. Shaina's locker code is divisible by all integers 2 through 6.
II. Shaina's locker code is divisible by 9 and 11.



[spoiler]made up by Sanjeev K Saxena for Avenues Abroad[/spoiler]

With statement 1, we know that Shaina's locker code is even and divisible by 5, so B must be 0. And since the code is divisible by 3, the sum of the digits must be divisible by 3.

Okay. So, the sum of the known digits is:

5 + 4 + 8 + 0 = 17

So, we will need to add 1, 4 or 7 to get a multiple of 3. (Anything else is a non-digit.) A could thus be 1, 4 or 7. Statement 1 is not sufficient.

With statement 2, the sum of the digits has to be a multiple of 9. The sum of the known digits is 5 + 4 + 8 = 17, so we'd need to add either 1 or 10 (anything else would involve adding a non-digit for either A or B) to get a multiple of 9.

We are left with a rather limited range of possible locker codes:

51480
50481

51488
52487
53486
54485
55484
56483
57482
58481

Which one(s) of these is a multiple of 11? The way we check is to add up the odd numbered digits and add up the even numbered digits, then subtract the first by the second and see whether we get a multiple of 11.

51480: 5 + 4 + 0 - (1 + 8) = 9 - 9 = 0 yes
50481: 5 + 4 + 1 - (0 + 8) = 10 - 8 = 2 no

51488: 5 + 4 + 8 - (1 + 8) = 17 - 9 = 8 no
52487: 5 + 4 + 7 - (2 + 8) = 16 - 10 = 6 no
53486: 5 + 4 + 6 - (3 + 8) = 15 - 11 = 4 no
54485: 5 + 4 + 5 - (4 + 8) = 14 - 12 = 2 no
55484: 5 + 4 + 4 - (5 + 8) = 13 - 13 = 0 yes
56483: 5 + 4 + 3 - (6 + 8) = 12 - 14 = 2 no
57482: 5 + 4 + 2 - (7 + 8) = 11 - 15 = 4 no
58481: 5 + 4 + 1 - (8 + 8) = 10 - 16 = 6 no

Our combination could thus be 51480 or 55484 (I doubt if it's divisible by 9, moreover it's a five-distinct-digit locker code, hence this possibility doesn't exist). Statement 2 is not sufficient.

Statements 1 and 2 combined only give us the possibility of 51480. Thus A = 1, and the two statements combined are sufficient. The correct answer would be C.

On a side note, be extremely careful about problems that you get from somewhere other than the official guide. I have never seen a real GMAT question that played off of the algorithm for multiples of 11 for numbers larger than 250 unless you could tell plainly from the digits (i.e., 1,100,110) as it takes FAR, FAR too long to realistically expect even a very bright GMAT student to get done in a reasonable amount of time.

Holy cow!

I did never imagine it could go so cumbersome, while dealing with the statement II. If the locker code 5A48B is divisible by 11. Then the difference in (5 + 4 + B) and (A + 8) is 0 or 11 or a bigger multiple of 11 (bigger multiples are although impossible in case of 5-digit numbers).

The difference (A + 8) - (5 + 4 + B) = A - B - 1, just ask yourselves; can A - B - 1 = 11 or more? Or can A - B = 12 or more? I'd say no! Because the difference in no two digits can be that big. Hence, A - B - 1 = 0; or A - B = 1, which means that A and B are consecutive digits where A > B.

Moreover, if the locker code 5A48B is also divisible by 9, then A + B + 17 is also divisible by 9 or A + B + 8 is also divisible by 9. Bearing in mind that A and B are consecutive digits also, where A > B; A + B can be 1, cannot be 10, forget 19 or more. Simply say that A = 1 and B = 0. Sufficient

[spoiler]IMO B[/spoiler]