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If r and s are positive integers, is rs a multiple of 25?

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If r and s are positive integers, is rs a multiple of 25?  [#permalink]

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New post Updated on: 18 Jan 2016, 04:18
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If r and s are positive integers, is rs a multiple of 25?

1) r-s is a multiple of 5
2) rs is a multiple of 5


*A solution will be posted in two days.

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Originally posted by MathRevolution on 13 Jan 2016, 19:13.
Last edited by Bunuel on 18 Jan 2016, 04:18, edited 1 time in total.
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Re: If r and s are positive integers, is rs a multiple of 25?  [#permalink]

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New post 13 Jan 2016, 20:21
MathRevolution wrote:
If r and s are positive integers, is rs a multiple of 25?

1) r-s is a multiple of 5
2) rs is a multiple of 5


*A solution will be posted in two days.


[I]: if r = 5a + k and s = 5b + k => r - s = 5 x (a-b) => a multiple of 5; but rs = (5a +1) (5b+1) = 5(5ab + a + b) +1 => rs not a multiple of 25
if r = 5a and s = 5b => rs = 25ab, a multiple of 25

===> [I] alone is not sufficient

[II]: if rs is a multiple of 5, for this case, we can make sure that there is at least one number, r or s, to be divisible by 5, it means that r = 5a or/and s=5a
But it is clearly not sufficient to know whether rs is a multiple of 25 with this [II] alone.

[I] + [II] : when combine, the data is likely to:
(1) if r=5a then it must be: s = 5b OR if r=5a + k, then it must be: s=5b+k
(2) at least r=5a or s =5b
==> [I] + [II] => r=5a and s=5b ==> rs=25ab, a multiple of 25. ==> Sufficient


Answer is C
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Re: If r and s are positive integers, is rs a multiple of 25?  [#permalink]

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New post 18 Jan 2016, 04:04
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If r and s are positive integers, is rs a multiple of 25?
1) r-s is a multiple of 5
2) rs is a multiple of 5


In the original condition, there are 2 variables(r,x), which should match with the number of equations. So you need 2 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer. When 1) & 2), a multiple of rs=5 -> a multiple of r=5 -> a multiple of r-s=5 -> a multiple of s=5 -> a multiple of rs=5*a multiple of 5=a multiple of 25, which is yes.
A multiple of s=5 -> a multiple of r-s=5 -> a multiple of r=5 -> a multiple of rs=5* a multiple of 5=a multiple of 25, which is yes. Therefore, the answer is C.


 For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: If r and s are positive integers, is rs a multiple of 25?  [#permalink]

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New post 16 Mar 2016, 07:02
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Re: If r and s are positive integers, is rs a multiple of 25?  [#permalink]

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New post 28 Feb 2019, 23:16
Hi

Here are my two cents for this question

We are asked if product of rs if mulitple of 25k

so if rs =25k,
this can be possible in following scenarios
(a) if r is multiple of 25 or
(b)s is multiple of 25 or
(c)r and s are each multiple of 25
(d)or r and s are each multiples of 5.

Now statement 1 gives us that difference of two integers is multiple of 5

Essentially they could be each not multiple of 5 or 25 and we could have difference which is multiple of 5 or we could have each multiple of 5 or 25 and we could have difference which is multiple of 5

here is what i mean .
if r= 21, s=11 r-s = 10 which is multiple of 5 or we can have r= 30 s =5 difference is 25 which is multiple of 5

From statement 2 we have either r or s is a multiple of 5

Now combining both we have

r or S is multiple of 5,

then
if r is multiple of 5 then s needs to be multiple of 5. because multiples of 5 can have either 0 or 5 in its units place . so if r takes 0 or 5 in its units place s also needs to take 0 or 5 in its unit place. Which makes r and s both multiples of 5

Hence both statements are required to answer the question .

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Re: If r and s are positive integers, is rs a multiple of 25?   [#permalink] 28 Feb 2019, 23:16
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