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# What is the three-digit number abc, given that a, b, and c

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What is the three-digit number abc, given that a, b, and c [#permalink]  15 Mar 2011, 22:45
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37% (02:23) correct 62% (01:35) wrong based on 1 sessions
What is the three-digit number abc, given that a, b, and c are the positive single digits that make up the number?

(1) a = 1.5b and b = 1.5c
(2) a = 1.5x + b and b = x + c, where x represents a positive single digit
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.  (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) Each statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.
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Re: three-digit number [#permalink]  15 Mar 2011, 23:35
Onell wrote:
What is the three-digit number abc, given that a, b, and c are the positive single digits that make up the number?

(1) a = 1.5b and b = 1.5c
(2) a = 1.5x + b and b = x + c, where x represents a positive single digit
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.  (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) Each statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

Statement 1 gives a=1.5*b and b=1.5*c and hence a = 2.25*c

Since a,b and c are integers, a has to be an integer that yields integral values when divided by 1.5 as well as 2.25, so it can only be 9 and hence a and c can be uniquely calculated as well, so sufficient.

Statement 2 says a = 1.5x + b and b = x + c and hence a = 2.5x + c

Since a is an integer, x can take only one value, i.e. 2 to yield a = 5+c and b = 2+c, but c can take multiple values (1,2,3 or 4, it cant be more than 4 as that will make b greater than 9) and hence insufficient.

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Re: three-digit number [#permalink]  16 Mar 2011, 00:01
Out of all + single digits - only even ones can be multiplied by 1.5 to yield a +ve single digit --> b and c have to be from amongst 2,4,6,8. and c has to be smallest.
If c=2, b=3, a=4.5-- wrong
If c=4, b=6, a=9-- maybe
If c=6, b=9, a will not be a single digit, or an integer!
No need of trying with 8, as 6 itself was too large to satisfy.
Thus abc can be determined as a single value : 964
Statement 1is sufficient

2. Just as above, 1.5 x has to yield a single digit, for it to add to another single digit (b) to yield single digit a. x has to be even
If x=2, a=3+b, b=2+c
If x=4, a=6+b, b=4+c-- but this is not possible-- even if x is minimum value 0, b is 4, and a is 10.
Thus no need to check with higher values.
But a=3+b, and b = 2+c can yield multiple calues - abc can be 631, 742, 853, 964
Since Statement 2 doesnt yield a single value, it is not sufficient.
Answer option A. 1 alone is sufficient.
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Re: three-digit number [#permalink]  16 Mar 2011, 22:59
From (1)

a = 3/2b and b = 3/2c

=> a = 9/4C

Hence a is a multiple of 9 it has to be 9 (because it's a single digit)

Hence c = 4, and b = 6

From(2), a = 1.5x + b and b = x + c

=> a = 2.5x + c

=> a-c = 2.5x = 5x/2

Now 2.5x must be an single digit integer, so x = 2 and a-c = 5

So a = b + 3 and b = 2 + c

but if c = 1 b = 3 a = 6
c = 2 b = 4 and a = 7, so not sufficient.

So the answer is A.
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Re: three-digit number   [#permalink] 16 Mar 2011, 22:59
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