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
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If a, b and c are single digit numbers
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GMATGuruNY
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For an integer to be divisible by 9, the sum of its digits must be a multiple of 9.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.
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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Jay@ManhattanReview
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Statement 1: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
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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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.
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.
