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# What is the remainder when a 3 digit integer ABC where A,B,C are its h

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Schools: Boston U '20 (M)
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What is the remainder when a 3 digit integer ABC where A,B,C are its h  [#permalink]

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23 Apr 2016, 06:54
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Difficulty:

75% (hard)

Question Stats:

47% (02:06) correct 53% (01:53) wrong based on 57 sessions

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What is the remainder when a 3 digit integer ABC where A,B,C are its hundreds,tens and units digits respectively is divided by 7?
[A] 2A+3B +C is divisible by 7
[B] The three Digit number is divisible by 49.

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What is the remainder when a 3 digit integer ABC where A,B,C are its h  [#permalink]

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23 Apr 2016, 21:24
Is ABC divisible by 7?

St1: 2A + 3B + C is divisible by 7
A can take values from 1 to 9, B from 0 to 9 and C from 0 to 9
Lets consider one value of 2A + 3B + C that is divisible by 7 = 2(1) + 3(0) + 5 = 7 --> Value of ABC = 105 --> divisible by 7
Now consider the next 6 values: 106, 107, 108, 109, 110, 111 --> 2A + 3B + C is not divisible by 7
Take 112 which is divisible by 7 --> 2A + 3B + C is divisible by 7
We observe a pattern here and 2A + 3B + C will be divisible by 7 only if the number is divisible by 7
Sufficient

St2: 49 = 7 * 7 --> Since 7 is a factor of 49, if the number is divisible by 49, it is also divisible by 7
Sufficient

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Re: What is the remainder when a 3 digit integer ABC where A,B,C are its h  [#permalink]

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23 Apr 2016, 23:20
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stonecold wrote:
What is the remainder when a 3 digit integer ABC where A,B,C are its hundreds,tens and units digits respectively is divided by 7?
[A] 2A+3B +C is divisible by 7
[B] The three Digit number is divisible by 49.

Ans

Statement 1 :

Write ABC as 100A +10B+C, or it can also be written as 50 ( 2A+3B+C ) -(140B+49C )

Hence, ABC = 50 ( 2A+3B+C ) -(140B+49C ) = 50 ( 2A+3B+C ) - 7( 20B + 7C )

Now, it is given that ( 2A+3B+C ) a factor of 7 thus the whole term 50 ( 2A+3B+C ) - 7( 20B + 7C ) is a factor of 7 , or ABC is a factor of 7. Thus, statement 1 is sufficient.

Statement 2 :

If ABC is divisible by 49 , we can write ABC = 49 X, hence ABC should also be divisible by 7.

Statement 2 is also sufficient alone.

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Re: What is the remainder when a 3 digit integer ABC where A,B,C are its h  [#permalink]

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03 Jan 2019, 07:41
stonecold wrote:
What is the remainder when a 3 digit integer ABC where A,B,C are its hundreds,tens and units digits respectively is divided by 7?
[A] 2A+3B +C is divisible by 7
[B] The three Digit number is divisible by 49.

The three digit integer is 100A+10B+C.

A) 2A+3B +C is divisible by 7. We can write the three digit integer 100A+10B+C as (98A+7B) + (2A+3B+C). 98A+7B is always divisible by 7 and given (2A+3B+C) is divisible by 7. So the given three digit number is divisible by 7. This statement is sufficient.

B) 7 is a factor of 49. If a number is divisible by 49, it should be divisible by 7. So statement is sufficient.

Hence D.

Cheers!
Re: What is the remainder when a 3 digit integer ABC where A,B,C are its h   [#permalink] 03 Jan 2019, 07:41
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