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If a, b and c are single digit numbers

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Source: — Data Sufficiency |
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by GMATGuruNY » Tue Jun 20, 2017 5:20 am
aaron1981 wrote:If a, b and c are single digit numbers from 1 to 9, inclusive, is „(a + ‚b +‚ c) divisible by 9?

(1) The number 2ab3 is divisible by 9.
(2) The number 4bc1 is divisible by 9.
For an integer to be divisible by 9, the sum of its digits must be a multiple of 9.

Statement 1:
No information about c.
INSUFFICIENT.

Statement 2:
No information about a.
INSUFFICIENT.

Statements combined:
Case 1: b=1, a=3, and c=3, with the result that 2ab3 = 2313 and 4bc1 = 4131.
In this case, a+b+c = 3+1+3 = 7, so the answer to the question stem is NO.
Case 2: b=8, a=5, c=5, with the result that 2ab3 = 2583 and 4bc1 = 4851.
In this case, a+b+c = 5+8+5 = 18, so the answer to the question stem is YES.

Since the answer to the question stem is NO in Case 1 but YES in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.
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by Jay@ManhattanReview » Tue Jun 20, 2017 5:52 am
aaron1981 wrote:If a, b and c are single digit numbers from 1 to 9, inclusive, is „(a + ‚b +‚ c)… divisible by 9?

(1) The number 2ab3 is divisible by 9.
(2) The number 4bc1 is divisible by 9.

OA E
Statement 1:

Since 2ab3 is divisible by 9, the sum of digits, i.e. „(2 +‚ a +‚ b +‚ 3) =… ƒ(5 ‚+ a ‚+ b)… is divisible
by 9.

=ƒ> 5 + a ‚+ b ƒ = 9k, where k is a positive integer
ƒ=> a +‚ b ƒ= 9k - 5 . . . (i)

However, there is no information about c. - Insufficient

Statement 2:

Since 4bc1 is divisible by 9, the sum of digits, i.e. „(4 ‚+ b ‚+ c ‚+ 1)…= ƒ „(5 ‚+ b ‚+ c…) is divisible
by 9.

ƒ=> 5 + b +‚ c ƒ= 9m, where m is a positive integer
ƒ=> b +‚ c ƒ = 9m - 5 . . . (ii)

However, there is no information about a. - Insufficient

Statement 1 & 2 together:

Adding (i) and (ii):

a ‚+ 2b +‚ c ƒ= 9„(k ‚+ m) - 10
=ƒ> a +‚ b ‚+ c ƒ= 9„(k ‚+ m) - 10 - b

However, the value of b is not known:

* If b ƒ= 8: a + b ‚+ c ƒ = 9„(k ‚+ m) - 10 - 8… = 9„(k ‚+ m -2), which is divisible by 9.
*If b =ƒ 1: a +‚ b ‚+ c ƒ= 9„(k ‚+ m) - 10 - 1… = 9„(k ‚+ m ) - 11, which is not divisible by 9.

Thus, the answer cannot be uniquely determined. - Insufficient

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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by Matt@VeritasPrep » Thu Jun 22, 2017 10:44 pm
If an integer is divisible by 9, then the sum of its digits is divisible by 9, and vice versa.

S1::

5 + a + b = 9 * something

Since 0 < a + b ≤ 18, we can only have 5 + a + b = 9 or 5 + a + b = 18. That gives a + b = 4 or a + b = 13. Not sufficient.

S2::

Same logic, b + c = 4 or b + c = 13. Also not sufficient.

S1 + S2::

Let's assume b + c = 13. If a + b = 13 as well, then we could have a = 5, b = 8, c = 5, in which case the answer is yes. But if a + b = 4, then we might have a = 0, b = 4, c = 9, in which case the answer is no.
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