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Basic strategy - number properties

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Basic strategy - number properties

by lkcr » Thu Apr 05, 2012 7:05 pm
Wanted to know what approach(es) people would suggest to questions such as this one here:

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?

a)3
b)4
c)12
d)32
e)35

Thanks!
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by Anurag@Gurome » Thu Apr 05, 2012 7:19 pm
lkcr wrote:Wanted to know what approach(es) people would suggest to questions such as this one here:

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?

a)3
b)4
c)12
d)32
e)35

Thanks!
n = 5*a + 1.
n = 7*b + 3.
Here, a and b are integers.
Note that the difference between divider and remainder (5 - 1 and 7 - 3) is 4 in both the case.
So add 4 on both sides of each of the 2 equations.
So, we get n+4 = 5*a+5 = 5*(a+1).
n+4 = 7*b+7 = 7*(b+1).
This means n+4 is a multiple of both 5 and 7.
Since 5 and 7 are co-prime, n+4 has to be a multiple of 5*7 = 35 as well.
So the smallest possible value of k is 4.

The correct answer is B.
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by gmatmath » Thu Apr 05, 2012 7:20 pm
First, we have to find 'n'. Let us find 'n' by trial and error method.
1) n = 11,
so, n%5 = 1,n%7 = 4 [% means gives the remainder]
2) n = 16
so, n%5 = 1, n%7 = 2
3)n = 21
so, n%5 = 1, n%7 = 0
4)n = 26
so, n%5 = 1, n%7 = 5
5)n = 31
so, n%5 = 1, n%7 = 3

Hence, we will consider the 5th option as it satisfies the criteria. the number 'n' = 31.
We are given k + n = 35
Knowing n = 31, we get k = 4
Therefore, the correct answer is k = 4, which is option b.
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by dchhikara863 » Thu Apr 05, 2012 7:23 pm
Any easy approach would be to pick numbers for the algebraic values.

And in most problem solving questions, answer choices itself provides us with clues on what numbers to pick!

In this question we need the smallest positive integer value for k right, so that k+n is a multiple of 35?

We can start by taking the small integers in the ans. 3,4,12
Start by option b.
Let's say K were = to 4, and k+n be 35(smallest multiple of 35 to keep it simple), so N is 31
Now, u can perform the algebraic steps that the questions requires.

31/5 gives 1 as the remainder. and 31/7 gives 3 as the remainder. That's the answer.
If the number we picked hadn't yield the right answer, we could have checked for 3 and 12 in the same way.

It's very unlikely that 32 and 35 are the answers as GMAT ques. rarely over-trick you :)
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by GMATGuruNY » Thu Apr 05, 2012 8:35 pm
I posted a solution here:

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by amit28it » Fri Apr 06, 2012 8:57 pm
gmatmath wrote:First, we have to find 'n'. Let us find 'n' by trial and error method.
1) n = 11,
so, n%5 = 1,n%7 = 4 [% means gives the remainder]
2) n = 16
so, n%5 = 1, n%7 = 2
3)n = 21
so, n%5 = 1, n%7 = 0
4)n = 26
so, n%5 = 1, n%7 = 5
5)n = 31
so, n%5 = 1, n%7 = 3

Hence, we will consider the 5th option as it satisfies the criteria. the number 'n' = 31.
We are given k + n = 35
Knowing n = 31, we get k = 4
Therefore, the correct answer is k = 4, which is option b.
The answer given by you is absolutely correct and I think your explanation is very simple and easy in comparison to others although there answers is also correct.
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by Shalabh's Quants » Fri Apr 06, 2012 9:33 pm
As per the I condn., n is an element of set {6,11,16,21,26,31,36....}.

Similarly, As per the II condn., n is also an element of set {10,17,24,31,38....}.

Smallest value of n satisfying both condns. is '31'.

As K+n should be 35(Being smallest).K+n=35 => K+31=35=> K=4.
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