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# If a and b are two-digit positive integers greater than 10

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Manager
Joined: 21 Jul 2018
Posts: 72
WE: Operations (Consulting)
Re: If a and b are two-digit positive integers greater than 10  [#permalink]

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17 Dec 2018, 18:37
Bunuel wrote:
SOLUTION

If $$a$$ and $$b$$ are two-digit positive integers greater than 10, is the remainder when $$a$$ is divided by 11 less than the remainder when $$b$$ is divided by 11?

First of all, note that the remainder when a positive integer is divided by 11 could be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10.

(1) The remainder when $$a$$ is divided by 69 is the fifth power of a prime number:

$$a=69q+prime^5$$ and $$prime^5<69$$ (the remainder must be less than the divisor). The only prime number whose fifth power is less than 69 is 2: $$(2^5=32) < 69$$. So, we have that $$a=69q+32$$: 32, 101, 170, ... Since $$a$$ is a two-digit integer, then $$a=32$$.

Now, $$a=32$$ divided by 11 gives the remainder of 10. Since 10 is the maximum remainder possible when divided by 11, then this remainder cannot be less then the remainder when $$b$$ is divided by 11 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10). So, the answer to the question is NO.

Sufficient.

(2) The remainder when $$b$$ is divided by 12 is $$b$$:

The remainder must be less than the divisor, hence $$b$$ must be less than 12 and since we are told that $$b$$ is a two-digit integers greater than 10, then $$b$$ must be 11.

Now, $$b=11$$ divided by 11 gives the remainder of 0. Since 0 is the minimum remainder possible, than the remainder when $$a$$ is divided by 11 cannot possible be less than 0. So, the answer to the question is NO.

Sufficient.

Try NEW remainders PS question.

Hi Bunuel,

I don't think option 2 is alone sufficient as they have not provided in question that a and b are two different integers, As per option 2 I agree that b must be 11 but what a is also valued as 11.

Correct me if I am missing something.

gmatbusters Could you please also review above mention doubt of mine and advise.
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Re: If a and b are two-digit positive integers greater than 10  [#permalink]

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17 Dec 2018, 19:01
1
Hi Gmatprep550

As per statement 2, b is 11. Hence the remainder when b is divided by 11 is 0.

Now 0 is the minimum remainder possible, a cannot give remainder lower than 0.

Your doubt: even if both a and b are equal.
Both remainders would be same = 0.

We can definitely say remainder when a is divided by 11 can not be lower than remainder when b is divided by 11

Hope, it is clear now. Don't hesitate to tag me again .

Gmatprep550 wrote:
Bunuel wrote:
SOLUTION

If $$a$$ and $$b$$ are two-digit positive integers greater than 10, is the remainder when $$a$$ is divided by 11 less than the remainder when $$b$$ is divided by 11?

First of all, note that the remainder when a positive integer is divided by 11 could be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10.

(1) The remainder when $$a$$ is divided by 69 is the fifth power of a prime number:

$$a=69q+prime^5$$ and $$prime^5<69$$ (the remainder must be less than the divisor). The only prime number whose fifth power is less than 69 is 2: $$(2^5=32) < 69$$. So, we have that $$a=69q+32$$: 32, 101, 170, ... Since $$a$$ is a two-digit integer, then $$a=32$$.

Now, $$a=32$$ divided by 11 gives the remainder of 10. Since 10 is the maximum remainder possible when divided by 11, then this remainder cannot be less then the remainder when $$b$$ is divided by 11 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10). So, the answer to the question is NO.

Sufficient.

(2) The remainder when $$b$$ is divided by 12 is $$b$$:

The remainder must be less than the divisor, hence $$b$$ must be less than 12 and since we are told that $$b$$ is a two-digit integers greater than 10, then $$b$$ must be 11.

Now, $$b=11$$ divided by 11 gives the remainder of 0. Since 0 is the minimum remainder possible, than the remainder when $$a$$ is divided by 11 cannot possible be less than 0. So, the answer to the question is NO.

Sufficient.

Try NEW remainders PS question.

Hi Bunuel,

I don't think option 2 is alone sufficient as they have not provided in question that a and b are two different integers, As per option 2 I agree that b must be 11 but what a is also valued as 11.

Correct me if I am missing something.

gmatbusters Could you please also review above mention doubt of mine and advise.

Posted from my mobile device
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Manager
Joined: 21 Jul 2018
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If a and b are two-digit positive integers greater than 10  [#permalink]

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18 Dec 2018, 07:11
Hi gmatbusters,

Thanks for quick response, I guess I got your point and wanted to make sure what I understood is correct.

As per your below mentioned statement what I am getting is remainder when a is divided by 11 can not be lower than remainder when b is divided by 11 as it's going to be equal or more.

"We can definitely say remainder when a is divided by 11 can not be lower than remainder when b is divided by 11"

_________________

______________________________
Consider KUDOS if my post helped !!

I'd appreciate learning about the grammatical errors in my posts

Please let me know if I'm wrong somewhere

Senior DS Moderator
Joined: 27 Oct 2017
Posts: 1195
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
Re: If a and b are two-digit positive integers greater than 10  [#permalink]

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18 Dec 2018, 07:13
1
You got it Right.

Gmatprep550 wrote:
Hi gmatbusters,

Thanks for quick response, I guess I got your point and wanted to make sure what I understood is correct.

As per your below mentioned statement what I am getting is remainder when a is divided by 11 can not be lower than remainder when b is divided by 11 as it's going to be equal or more.

"We can definitely say remainder when a is divided by 11 can not be lower than remainder when b is divided by 11"

_________________
Manager
Joined: 21 Jul 2018
Posts: 72
WE: Operations (Consulting)
Re: If a and b are two-digit positive integers greater than 10  [#permalink]

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18 Dec 2018, 07:19
1
Thanks again for quick response, you rock !!

gmatbusters wrote:
You got it Right.

Gmatprep550 wrote:
Hi gmatbusters,

Thanks for quick response, I guess I got your point and wanted to make sure what I understood is correct.

As per your below mentioned statement what I am getting is remainder when a is divided by 11 can not be lower than remainder when b is divided by 11 as it's going to be equal or more.

"We can definitely say remainder when a is divided by 11 can not be lower than remainder when b is divided by 11"

_________________

______________________________
Consider KUDOS if my post helped !!

I'd appreciate learning about the grammatical errors in my posts

Please let me know if I'm wrong somewhere

Re: If a and b are two-digit positive integers greater than 10 &nbs [#permalink] 18 Dec 2018, 07:19

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