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AB4AC If teh 5 digit number above is divisible by 45, and A,

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Director
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AB4AC If teh 5 digit number above is divisible by 45, and A, [#permalink]

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05 Oct 2004, 18:57
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AB4AC
If teh 5 digit number above is divisible by 45, and A, B, and C are distinct digits, what is the value of the number?
I) A + C = 3
II) A * C = 0 and A * B = 24
Director
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05 Oct 2004, 20:00
D for me.

AB4AC is divisible by 45 if it can be divisible by both 9 and 5.

Divisibility of 9:sum of digits is 9.
Divisibility of 5:last digit is 0 or 5.

1. We have A+C =0.Possibilities are A=0 or 1 or 2 or 3, C=3 or 2 or 1 or 0. Since the last digit is C, the only option we have is C=0 (for the number to be divisible by 5). So, A=3

We then have the number as 3B430. For the number to be divisible by 9, B has to be 8. Only number. hence sufficient.

2.A*C=0, AB=24. combining these two we can say that A cannot be 0. So, C should be 0. Now, forA and B the possibilities are A=3,6,4 or 8, B =8,4,6 or 3. but 4 is not a possibility because there is already a 4 in the 5 digit number and we have the numbers are distinct. this also means 6 is not possible, because if 6 there has to be a 4.

Only possibility is 3and 8 for either A or B. However, if A is 8 then the number does not fit divisibility of 9. Hence only possibility is A=3 andB=8. Hence sufficient.

Ans is D.
Manager
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05 Oct 2004, 20:24
You hit the hammer right at the nail
Senior Manager
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05 Oct 2004, 23:48
Got D. Same as venksune.
What a time consumer..
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06 Oct 2004, 04:13
Hi Venksune,

I do not understand your point when you write divisible by 9 <=> sum of digits = 9

One more point : I think just A, B and C have to be distinct integers, I am not so sure 4 can not be included in the choices.
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06 Oct 2004, 13:29
Quote:
If teh 5 digit number above is divisible by 45, and A, B, and C are distinct digits, what is the value of the number?

I agree with Twixt. It looks as though 4 is a viable option for either A, B, or C. Therefore the answer is A because in statement (2) A can be either 8 or 4.

Quote:
I do not understand your point when you write divisible by 9 <=> sum of digits = 9

Twixt - This is a number properties rule, for every multiple of 9, the sum of the digits equals a multiple of 9. 54 -> 5+4=9, 108 ->1+0+8 = 9, 198 -> 1+9+8 = 18. I'm not quite sure of the mathematical proof as to why this is true, it just is. Same goes for 3...for every multiple of 3, the sum of the digits equals a multiple of 3.
Director
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06 Oct 2004, 13:52
Thank you Sofere,

I thought about this during my transportation today and I noticed It is true but I did not find any obvious reason !

So being a property of 9 this digits sum is a property of 3. I never noticed this one in any book...

If I try to summarize :

divisible by 2 : last digit = 0,2,4,6,8
3 : sum of digits is a multiple of 3
4 : last digit = 2 times 2 rule ?
5 : last digit = 0,5
6 : mix of 2 and 3
7 : do not know
8 : last digit = 3 times 2 rule ?
9 : sum of digits is a multiple of 9

Any volunteer to fill in the blanks ?
Director
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06 Oct 2004, 14:17
So BEFORE being a property of 9 this digits sum is a property of 3
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06 Oct 2004, 14:22
twixt wrote:
Thank you Sofere,

I thought about this during my transportation today and I noticed It is true but I did not find any obvious reason !

So being a property of 9 this digits sum is a property of 3. I never noticed this one in any book...

If I try to summarize :

divisible by 2 : last digit = 0,2,4,6,8
3 : sum of digits is a multiple of 3
4 : last digit = 2 times 2 rule ?
5 : last digit = 0,5
6 : mix of 2 and 3
7 : do not know
8 : last digit = 3 times 2 rule ?
9 : sum of digits is a multiple of 9

Any volunteer to fill in the blanks ?

Here you go...

divisible by
2 : last digit = even
3 : sum of digits is a multiple of 3
4 : last two digits is a multiple of 4
5 : last digit = 0,5
6 : mix of 2 and 3 (last digit is even and sum is divisible by 3)
9 : sum of digits is a multiple of 9
10: last digit is 0
12: mix of 3 and 4 (last two digits are divisible by 4 and sum is divisible by 3)
Director
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06 Oct 2004, 14:31
sofere wrote:
Quote:
If teh 5 digit number above is divisible by 45, and A, B, and C are distinct digits, what is the value of the number?

I agree with Twixt. It looks as though 4 is a viable option for either A, B, or C. Therefore the answer is A because in statement (2) A can be either 8 or 4.

OA is A.
Manager
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06 Oct 2004, 14:58
Quote:
divisible by 2 : last digit = 0,2,4,6,8
3 : sum of digits is a multiple of 3
4 : last digit = 2 times 2 rule ?
5 : last digit = 0,5
6 : mix of 2 and 3
7 : do not know
8 : last digit = 3 times 2 rule ?
9 : sum of digits is a multiple of 9

I'm not familiar with the last digit = 2 times 2 rule or 3 times 2 rule. For 4s and 8s i see if the entire # is divisble by 2 twice or 3 times respectively. Going by the last digit doesn't work for 4s and 8s. Also if the last 2 digits are divisible by 4, the number is divisible by 4 (i.e. 1028...28 is divisible by 4 so 1028 is too).

Also for 100-999...if the first and last digit add up to the middle digit, the number is divisible by 11.
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06 Oct 2004, 18:34
OnlyABC are distinct.
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06 Oct 2004, 23:07
from The first one
(1) A=3,C=0 , B=8 . The number is 38430. as above so the first information can solve our problem

for the second. we have A*C=0 A*B=24

This gives C=0, A=4 or B=6 and vice versa (46440 not divisible by 45)
A=3 or B=8 and vice versa (38430 divisible by 45

Director
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07 Oct 2004, 06:41
Thank you Hardworker for your help.

I did not find any trick for 7, for 8 we can say that last 3 digits are multiple of 8 !
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07 Oct 2004, 22:52
looking at number 45 suggests last digit of AB4AC shud be 0 or 5 in any case.

so ,working with that on first condition
a> A=3,C=0 and B=8

and with second case given we can easily dedudce
b>A=3,C=0,B=8

so both ways we can arrive at solution..
hence,A

-cheers
vivek
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07 Oct 2004, 23:02
sorry,i don't know which option has wat notations,i mean abt. A,B,C,DorE.

so as this problem can be solved by both cases independently,then can anyone let me know what will be option of answer..

Please let me know where i can see the meaning of options..
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08 Oct 2004, 18:07
I assume vivek.173 you are new to data suficiency questions. They have a special format which can be found in any gmat book.

choice A : statement 1 alone is sufficient but statement 2 alone is
insufficient.

A, it seems is the correct answer (based on many earlier posts)

statement 2 alone is insufficient.
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