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If a and b are positive integers, what is the remainder when ab is

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kingflo
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If a and b are positive integers, what is the remainder when ab is [#permalink] New post   Updated on: 18 Nov 2017, 02:07
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If a and b are positive integers, what is the remainder when ab is divided by 40?

(1) b is 60% greater than a.

(2) Each of \(a^2b\) and \(ab^2\) is divisible by 40.
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A

Originally posted by kingflo on 20 Jul 2013, 09:26.
Last edited by Bunuel on 18 Nov 2017, 02:07, edited 2 times in total.
Edited the question.
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   20 Jul 2013, 09:35
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If a and b are positive integers, what is the remainder when ab is divided by 40?

(1) b is 60% greater than a --> \(b=1.6a\)--> \(b=\frac{8}{5}a\) --> \(\frac{b}{a}=\frac{8}{5}\) --> b is a multiple of 8 (8x) and a is a multiple of 5 (5x) --> ab=5x*8x=40x^2. The remainder when \(ab=40x^2\) is divided by 40 is 0. Sufficient.

(2) Each of a^2 *b and a*b^2 is divisible by 40. If a=2 and b=10, then ab=20 and the the remainder is also 20 but if a=b=40, then ab=40^2 and the remainder si 0. Not sufficient.

Answer: A.
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kingflo
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   20 Jul 2013, 09:39
Could we also pick numbers for (1)?

(1) b = 1,6a
--> pick smart numbers: b=10 , a=16 --> 10*16 / 40 = 4 + R0 --> Sufficient

Right?
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   20 Jul 2013, 09:47
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kingflo wrote:
Could we also pick numbers for (1)?

(1) b = 1,6a
--> pick smart numbers: b=10 , a=16 --> 10*16 / 40 = 4 + R0 --> Sufficient

Right?


This won't work in all cases. If the question were "what is the remainder when ab is divided by 80?", then with your numbers you'd still get that the remainder is 0, and the statement is sufficient. But this would be wrong: if b=5 and a=5, then ab=40 and the remainder is 40 not 0.

Hope it's clear.
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   08 Sep 2015, 18:37
Hi Bunuel,
i got this question wrong,but can you poiunt out th eflaw in my reasoning here,
what i did was i calculated that b= 8* a /5 and then ab = (8 * a^2 )/5. now if i divide this new ab by 40 i am left with a^2/25. after this i plugged numbers for a and got different remainders i.e if a^2 = 1, 4, 9, 16 etc , i will have different remainders therefore S1 is insufficient.
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   08 Sep 2015, 21:37
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Jerry1982 wrote:
Hi Bunuel,
i got this question wrong,but can you poiunt out th eflaw in my reasoning here,
what i did was i calculated that b= 8* a /5 and then ab = (8 * a^2 )/5. now if i divide this new ab by 40 i am left with a^2/25. after this i plugged numbers for a and got different remainders i.e if a^2 = 1, 4, 9, 16 etc , i will have different remainders therefore S1 is insufficient.


From b= 8* a /5 (b/a = 8/5) it follows that a must be a multiple of 5, so you should plug only multiples of 5 there.
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   10 Sep 2015, 11:43
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Bunuel wrote:
kingflo wrote:
Could we also pick numbers for (1)?

(1) b = 1,6a
--> pick smart numbers: b=10 , a=16 --> 10*16 / 40 = 4 + R0 --> Sufficient

Right?


This won't work in all cases. If the question were "what is the remainder when ab is divided by 80?", then with your numbers you'd still get that the remainder is 0, and the statement is sufficient. But this would be wrong: if b=5 and a=5, then ab=40 and the remainder is 40 not 0.

Hope it's clear.


Isn't it possible to actually pick numbers here for (1) ?
We know both 'a' and 'b' have to be positive integers, and b is always 1.6 times of a.
So, all integer combinations which satisfy the condition - b=1.6a can help us.

    a --> b = 1.6 a
    5 --> 8
    10 --> 16
    15 --> 24
and likewise, and in all cases remainder will be 0. Right ?
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   26 Mar 2019, 17:11
You can pick numbers as long as you make sure to test a few combinations, what Bunuel was saying was that if the divisor was 80 we'd get R40 using the numbers 8*5 while we would get R0 using the next set of multiples, 16*10. So you can't just pick one set and assume it will be sufficient in remainder questions since there's often some pattern with different remainders, just not here.
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   19 May 2021, 03:37
Can someone pls tell me where am I going wrong with the second statement?
2) Each of \(a^2b\) and \(ab^2\) is divisible by 40.

That is:
\(a^2 b = 40x \) and
\(ab^2 = 40y \)
Subtracting the two equations
\(a^2 b - ab^2 = 40x - 40y \)
\(ab(a - b) = 40 ( x - y) \)
\(ab = \frac{[40( x- y)]}{(a-b)}\)

This tells us ab is a multiple of 40
So if ab is divided by 40, the remainder would be 0

Insights would be appreciated.

Thanks!
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   24 May 2021, 11:30
sm1510 wrote:
Can someone pls tell me where am I going wrong with the second statement?
2) Each of \(a^2b\) and \(ab^2\) is divisible by 40.

That is:
\(a^2 b = 40x \) and
\(ab^2 = 40y \)
Subtracting the two equations
\(a^2 b - ab^2 = 40x - 40y \)
\(ab(a - b) = 40 ( x - y) \)
\(ab = \frac{[40( x- y)]}{(a-b)}\)

This tells us ab is a multiple of 40
So if ab is divided by 40, the remainder would be 0

Insights would be appreciated.

Thanks!


The equation is kind of flawed here, what happens if a=b? you are essentially dividing by 0 which is undefined.
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   29 May 2021, 16:51
Do you mean we are not allowed to subtract the two equations ?

Posted from my mobile device
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   31 Mar 2022, 03:47
Bunuel wrote:
If a and b are positive integers, what is the remainder when ab is divided by 40?

(1) b is 60% greater than a --> \(b=1.6a\)--> \(b=\frac{8}{5}a\) --> \(\frac{b}{a}=\frac{8}{5}\) --> b is a multiple of 8 (8x) and a is a multiple of 5 (5x) --> ab=5x*8x=40x^2. The remainder when \(ab=40x^2\) is divided by 40 is 0. Sufficient.

(2) Each of a^2 *b and a*b^2 is divisible by 40. If a=2 and b=10, then ab=20 and the the remainder is also 20 but if a=b=40, then ab=40^2 and the remainder si 0. Not sufficient.

Answer: A.


Is there any other way of proving the insufficiency of Statement 2 ? Also how do we specifically choose these numbers? I did not get these numbers in my mind when I was trying this out.
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   13 Apr 2022, 18:04
sm1510 wrote:
Can someone pls tell me where am I going wrong with the second statement?
2) Each of \(a^2b\) and \(ab^2\) is divisible by 40.

That is:
\(a^2 b = 40x \) and
\(ab^2 = 40y \)
Subtracting the two equations
\(a^2 b - ab^2 = 40x - 40y \)
\(ab(a - b) = 40 ( x - y) \)
\(ab = \frac{[40( x- y)]}{(a-b)}\)

This tells us ab is a multiple of 40
So if ab is divided by 40, the remainder would be 0

Insights would be appreciated.

Thanks!


\(ab = \frac{[40( x- y)]}{(a-b)}\)

This does not necessarily mean ab is a multiple of 40. Observe you are dividing the Numerator (40K) by a-b. a-b could be a factor of 40, thus reducing the 40 in the numerator:

e.g. a=10, b=2; a-b=8;
therefore ab=5(x-y) -> Not always a multiple of 40
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink] New post   17 May 2023, 04:47
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Re: If a and b are positive integers, what is the remainder when ab is [#permalink]
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