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In the number 11,0AB, A and B represent the tens and units d

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n2739178
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In the number 11,0AB, A and B represent the tens and units d [#permalink] In the number 11,0AB, A and B represent the tens and units d   22 Jan 2011, 10:48
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In the number 11,0AB, A and B represent the tens and units digits, respectively. If 11,0AB is divisible by 55, what is the greatest possible value of B × A?

A. 0
B. 5
C. 10
D. 15
E. 25
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E
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Bunuel
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   22 Jan 2011, 11:04
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In the number 11,0AB, A and B represent the tens and units digits, respectively. If 11,0AB is divisible by 55, what is the greatest possible value of B × A?

A. 0
B. 5
C. 10
D. 15
E. 25

You should notice that 55*2=110 so 11,000 is divisible by 55: 55*200=11,000 (or you can notice that 11,000 is obviously divisible by both 5 and 11 so by 55) --> B*A=0*0=0. Next number divisible by 55 is 11,000+55=11,055: B*A=5*5=25 (next number won't have 110 as the first 3 digits so we have only two options 0 and 25).

Answer: E.


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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   22 Jan 2011, 10:53
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Hi all,

I came across this question on a CAT practice test....

I realised that for the number to be divisible by 55, it must be divisible by both 5 and 11...

However, the official explanation for the question is this:

"If 11,0AB is divisible by 55, it must be divisible by both 5 and 11. So, the only numbers it could be are 11,000 or 11,055. So A = 5 and B = 5, or A = 0 and B = 0. So the only possible values of B × A are 0 and 25. Which is greater? 25."

My question is, how would you know that "the only numbers it could be are 11,000 or 11,055" ? When I did this question under exam conditions, I tried to consider all increments of 5 between 11,000 and 11,099 (i.e. 11,005 ... 11,010 ... 11,015 etc.) and wasn't sure which one of those would be the highest number divisible by 55... I.e. for all I knew, 11,095 could have been divisible by 55 (i.e. 5 and 11)... What's a quicker way of doing this?

thanks
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   23 Jan 2011, 14:28
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n2739178
In the number 11,0AB, A and B represent the tens and units digits, respectively. If 11,0AB is divisible by 55, what is the greatest possible value of B × A?

A. 0
B. 5
C. 10
D. 15
E. 25

Another quick approach:

110AB is divisible by 5 and 11.
So B must be either 0 or 5. If B = 0, then BxA = 0 and is the minimum of the given options.
What if B = 5? The maximum option given is (E) i.e. 25 in which case A = 5. Now the question is: Is 11055 divisible by 11?
Divisibility rule of 11: Sum of odd place digits = 5 + 0 + 1 = 6
Sum of even place digits = 5 + 1 = 6
Since the difference in the sum is 0, 11055 is divisible by 55. So option (E) is indeed correct.
Note: Had we found that 11055 is not divisible by 11, we would have tried other options.
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   27 Jan 2011, 12:29
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HI ,

To add to the Answer : I think in such type of Questions reading the answer choices is as Important.

11000 is multiple of 55, we know the number has to be greater or equal to 11,000.

Option : 1 If its is 11,000 then B and A are 0'. And the choice is A (0)

Option : 2 If the Number is greater than 11,000 then it can be anything until 11,999.

Ex : 11,055 , 11110, 11165 ... but understand that the BiggestNumber among the choices given is - 25

So all we need to know if there is / are any numbers above 11,000 that are divisible by 55 and have the tens and Unit Digit whose product would be 25.

Also, Understand that 11165 is a Multiple and 6*5 = 30, but then its not among the Ans ChoIces.

Hope that Helps.
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   31 Jan 2011, 23:05
@davagabound the question is given as 110AB, so 11165 is automatically ruled out,

@ karishma, thanks again :) , please do provide some background for this method you used.

' Divisibility rule of 11: Sum of odd place digits = 5 + 0 + 1 = 6
Sum of even place digits = 5 + 1 = 6
Since the difference in the sum is 0, 11055 is divisible by 55 .'



@ bunuel , thanks.
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   01 Feb 2011, 03:12
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@davagabound the question is given as 110AB, so 11165 is automatically ruled out,

@ karishma, thanks again :) , please do provide some background for this method you used.

' Divisibility rule of 11: Sum of odd place digits = 5 + 0 + 1 = 6
Sum of even place digits = 5 + 1 = 6
Since the difference in the sum is 0, 11055 is divisible by 55 .'



@ bunuel , thanks.

Divisibility Rule for 11: If you sum every second digit and then subtract the sum of all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11.

Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11.

For more on Divisibility Rules check: math-number-theory-88376.html
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   01 Feb 2011, 17:11
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110AB -> The difference of the sums of alternate numbers must be 0 or divisible by 11 and also the number is multiple of 5. AB can 11 to 55, 55 being the max AB and 25 being the max A*B. E.
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   01 Feb 2011, 20:17
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garimavyas

@ karishma, thanks again :) , please do provide some background for this method you used.


Divisibility and Remainder Rule of 11:

When you have a number say 12345 and you want to find whether it is divisible by 11, do the following:
Start with the right most digit (5 here). Add to it every alternate digit i.e. 5 + 3 + 1 = 9 (These are the odd digits. 1st rightmost digit + 3rd rightmost digit etc)
Sum all even place digits 4 + 2 = 6
If these two sums differ by 0 or any multiple of 11, the number is divisible by 11.
Else it is not. In this case 9 - 6 = 3 so number is not divisible by 7.
Also, the remainder in this case is 3.
Remember, remainder is Sum of Odd digits - Sum of even digits
e.g. 12. Remainder here is 2 - 1 = 1
but 21 Remainder here is 1-2 = -1 i.e. 10

Similarly 12859
Odd digits sum = 9+8+1 = 18
Even digits sum = 5 + 2 = 7
Difference between the sums is 11 so this number is divisible by 11.
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   01 Feb 2011, 23:39
''davagabound the question is given as 110AB, so 11165 is automatically ruled out,''

Yeah I missed that ThankS.
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Re: Divisibility question [#permalink] In the number 11,0AB, A and B represent the tens and units d   02 Feb 2011, 02:27
@ karishma ,that clears the concept, thanks a lot ,thanks for all these methods

@ bunuel , thanks to you too , the number theory link is useful .
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