Bunuel wrote:

Let abc denote the digits of a three-digit number. Is this number divisible by 7?

(1) The two-digit number bc is divisible by 7

(2) a + b + c = S, and S is divisible by 7

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:Statement #1, by itself, doesn’t help us. The number 49 is divisible by 7, but think about 149, which equals (100 + 49): the divisor 7 would divide evenly into 49, but it wouldn’t divide evenly in to 100, so 149 can’t possibly be divisible by 7.

Another way to think about it: 140 is a multiple of 7, so 147 must be, and 154 must be as well. The number 149 falls between the multiples of 7. (In fact, you would not need to know this for the GMAT, but 149 is a prime number.)

Of course, if the hundred’s digit is 7, then all the number of this form would be divisible by 7: 735, 742, 749, 756, etc. So, we can get either a yes or a no answer with this statement.

This statement, alone and by itself, is not sufficient.

Statement #2: a + b + c = S, and S is divisible by 7

This is reminiscent of the trick for

divisibility by 3. That trick works for 3, but it doesn’t work for 7!!

Yes, we could find a number, such as 777, which is divisible by 7 and whose digits add up to 21, also divisible by 7. That number would produce a “yes” answer with this prompt.

But we also could find a number such as 223. The digits add up to 7, so this could be the number according to statement #2. Think about the multiples of 7 in that vicinity. Clearly, 21 is a multiple of 7, so 210 must also be. From there, 210, 217, 224, 231, etc. The number 223 is between the multiples of 7, so it not a multiple of 7. (In fact, you would not need to know this for the GMAT, but 223 is also a prime number.) This produces a “no” answer. Two different prompt answer possible.

This statement, alone and by itself, is not sufficient.

Combined statements:For our “yes” representative, we can pick 777: the last two digits form 77, a number divisible by 7; the sum of the digits is 21, divisible by 7; and the number 777 is most certainly divisible by 7.

For our “no” representative, we can pick 149: the last two digits form 49, a number divisible by 7; the sum of the digits is 149, divisible by 7; and, as discussed above, the number 149 is not divisible by 7.

Even with both statements, two prompt answers are possible. Even together, the statements are not sufficient.

Answer = (E)

(This is totally beyond what you would need to know for the GMAT, but the numbers 149 and 421 and 491 and 563 are the three digit numbers that satisfy both statements but are prime!!)
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