bhumika.k.shah wrote:a b c
+ d e f
= x y z

If, in the addition problem above, a, b, c, d, e, f, x, y, and z each represent different positive single digits, what is the value of z ?

(1) 3a = f = 6y

(2) f - c = 3

What can be the possible different approaches ??

start with f = 6Y. If f is a single digit number, it has to be 6 because the next multiple of 6 will violate the condition.

hence Y =1 , a = 2 and c = 3

therefore f + z = 9.

i tried solving it, but got stuck. statement 2 is clearly insufficient.

i'm not sure about A. i got the same values as harshavardhanc did: y = 1, a = 2 and c = 3. this is where i'm getting stuck!

rahul.s wrote:i tried solving it, but got stuck. statement 2 is clearly insufficient.

i'm not sure about A. i got the same values as harshavardhanc did: y = 1, a = 2 and c = 3. this is where i'm getting stuck!

Surely rahul.s,Alone the 2nd statement is insufficient.
But I guess ur soln. c=3 has come when you have considered both the statements together.
Now,

y = 1, a = 2 and c = 3. this is where i'm getting stuck!

Now,c+f = 9 which is equal to z .
Hence,the answer would be C.(Both together are sufficient]

The most imp. pt. to catch in the question is that "If f is a single digit number, it has to be 6 because the next multiple of 6 will violate the condition."

I hope it is clear now.

The OA is A

will provide the OE's after few discussions.

IMO A

s1)---->y=1,a=2 and f=6

now c+f=z----> for z to be a positive single unit c has to be 0 or 3 ( 1 and 2 already ocuupied) and for c>3 z= two digit no. and hence violate the given condition

now if c=0 z=f which is not possible since f and z are diff

therefore c is 3 and z is 9

suff

s2) insuff

thephoenix wrote:IMO A

s1)---->y=1,a=2 and f=6

now c+f=z----> for z to be a positive single unit c has to be 0 or 3 ( 1 and 2 already ocuupied) and for c>3 z= two digit no. and hence violate the given condition

now if c=0 z=f which is not possible since f and z are diff

therefore c is 3 and z is 9

suff

s2) insuff

Hey thephoenix,
But z can aslo take other values except 0 and 3.The tens digit would be carried over to the "addition of b + e".
So I don't think that any violation can take place:-

c>3 z= two digit no. and hence violate the given condition

I can give u an example:-
We know that f=6
Suppose c=7
Then,c + f =13 but "z" will only take the units' place which is "3".
The "1" in the tens place will be carried over for "addition of b+e".

We just have to make sure that "a + d +(Any carry) " should be less than 10 since we should get a single digit no.-x.

The problem states that all 9 single digits in the problem are different; in other words, there are no repeated digits.

(1) SUFFICIENT: Given 3a = f = 6y, the only possible value for y = 1. Any greater value for y, such as y = 2, would make f greater than 9. Since y = 1, we know that f = 6 and a = 2.

We can now rewrite the problem as follows:

2 b c
+ d e 6

= x 1 z

In order to determine the possible values for z in this scenario, we need to rewrite the problem using place values as follows:

200 + 10b + c + 100d + 10e + 6 = 100x + 10 + z

This can be simplified as follows:

196 = 100(x - d) - 10(b + e) + 1(z - c)

Since our focus is on the units digit, notice that the units digit on the left side of the equation is 6 and the units digit on the right side of the equation is (z - c). Thus, we know that 6 = z - c.

Since z and c are single positive digits, let's list the possible solutions to this equation.

z = 9 and c = 3
z = 8 and c = 2
z = 7 and c = 1

However, the second and third solutions are NOT possible because the problem states that each digit in the problem is different. The second solution can be eliminated because c cannot be 2 (since a is already 2). The third solution can be eliminated because c cannot be 1 (since y is already 1). Thus, the only possible solution is the first one, and so z must equal 9.

(2) INSUFFICIENT: The statement f - c = 3 yields possible values of z. For example f might be 7 and c might be 4. This would mean that z = 1. Alternatively, f might be 6 and c might be 3. This would mean that z = 9.

The correct answer is A.


hope the OE helps