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Re: If a and b are two-digit positive integers greater than 10
[#permalink]
17 Dec 2018, 19:37
17 Dec 2018, 19:37
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.
Answer: D.
Try NEW remainders PS question.
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.
Answer: D.
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.
GMAT Tutor
Joined: 27 Oct 2017
Posts: 1895
Given Kudos: 238
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Re: If a and b are two-digit positive integers greater than 10
[#permalink]
17 Dec 2018, 20:01
17 Dec 2018, 20:01
1
Kudos
Expert Reply
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 .
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
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.
Answer: D.
Try NEW remainders PS question.
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.
Answer: D.
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
Re: If a and b are two-digit positive integers greater than 10
[#permalink]
18 Dec 2018, 08:11
18 Dec 2018, 08:11
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"
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"
