# GMAT question 65

#### ironsheep

##### Junior Member
This is GMAT question 65 from the book, "The Official Guide for GMAT Quantitative Review 2017".

The question is that 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? The answer is 4.

I get that n= 7q +3 and it also equals n = 5p +1. I also know that you can make it to where 5p= 7q +2, but this is where I am really stuck as where do I go from there. I don't understand it when the book provides the explanation. I looked online and on videos and on those videos it said to find LCM, but why??

#### ironsheep

##### Junior Member
Can someone use image enhancement on this photo

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#### pka

##### Elite Member
This is GMAT question 65 from the book, "The Official Guide for GMAT Quantitative Review 2017".
The question is that 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? The answer is 4.
Just from the statement if the question, it is clear that n=31. If that is what is k if 31+k=35 ?

#### lev888

##### Full Member
The question is that 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? The answer is 4.

I get that n= 7q +3 and it also equals n = 5p +3.
Do you see how 5, 7 and 35 are related?
5p+3? Take another look.

#### ironsheep

##### Junior Member
I get the LCM of 7q and 5p is 35 as 7 times 5 is 35. Then you would end up with 35 equals 35 + 2 as 5p equals 7q + 2, I got that part, but the answer is 4 and I have a hard time understanding what the GMAT instructors wrote as explanation. I am still stuck.

#### pka

##### Elite Member
The question is that 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? The answer is 4.
I get the LCM of 7q and 5p is 35 as 7 times 5 is 35. Then you would end up with 35 equals 35 + 2 as 5p equals 7q + 2, I got that part, but the answer is 4 and I have a hard time understanding what the GMAT instructors wrote as explanation.
I think that you are demanding a way to solve and not seeing the point of the question. Just from the statement one should see that $$\displaystyle 0<n<35$$
$$\displaystyle 6,11,16,21,26,31$$ are the only positive integers, $$\displaystyle n<35$$, that have a remainder of $$\displaystyle 1$$ when divided by $$\displaystyle 5$$ of those which has a remainder of $$\displaystyle 3$$ when divided by $$\displaystyle 7~?$$ Moreover $$\displaystyle 31+4=35$$.
Among other things the GMAT testing reasoning skills.

#### lev888

##### Full Member
The question is that 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? The answer is 4.
I get that n= 7q +3 and it also equals n = 5p +1.
Ok, n = 7q + 3 and n = 5p + 1. We want to add something to n to get a number divisible by 35. Divisible by 35 means divisible by 5 and by 7.
Let's take 5 first. Which representation do we want to use when looking at divisibility by 5. Of course, 5p + 1. So what can we add so it's divisible by 5? That would be 4, 9, 14, etc.
Now 7. 7q + 3. What can we add so it's divisible by 7? 4, 11, 18, etc.
Looks like we don't need to go far to find the number present in both sequences - 4.

#### ironsheep

##### Junior Member
n= 7 times 2 plus 3 equals 17, but if you add four to it that means it will be divisible by 7 as 21 divided by 7 is 3. n= 5 times 4 plus 1 equals 21, which is divisible by 7, but not 5. Why would you need to add 4 to the n= 5p plus 1 then? Also, aren't we looking to use 35 or 70 or 105 and not 7 or 5??

#### lev888

##### Full Member
n= 7 times 2 plus 3 equals 17, but if you add four to it that means it will be divisible by 7 as 21 divided by 7 is 3. n= 5 times 4 plus 1 equals 21, which is divisible by 7, but not 5. Why would you need to add 4 to the n= 5p plus 1 then? Also, aren't we looking to use 35 or 70 or 105 and not 7 or 5??
n can't be 2 different numbers - 17 and 21, as you wrote. It's one number.
n = 7q + 3 and n = 5p + 1 - here q and p are such, that we get the same n from both expressions. 7*2+3 and 5*4+1 doesn't work. But my point is that we don't need to find what n is.

"k+n is a multiple of 35" means k+n is divisible by 35, no?
And what is 35? 35 = 5*7. If a number is divisible by 35 it's divisible by 5 and 7. So we are looking for n+k that is divisible by 5 and 7.

#### ironsheep

##### Junior Member
Are we looking for k? Isn't n important since it is part of the n +k equation? This problem is made confusing because the of the variables p and q, which the GMAT wrote in the answer explanation. Also, multiples of 35 are 35, 70, 105... etc

#### lev888

##### Full Member
Are we looking for k? Isn't n important since it is part of the n +k equation? This problem is made confusing because the of the variables p and q, which the GMAT wrote in the answer explanation. Also, multiples of 35 are 35, 70, 105... etc
We need to find k. Why is this a problem if we can do it without finding n?
For example, consider this problem: n is even, find smallest positive integer k such that n+k is even. Answer: k = 2. Doesn't matter what n is and whether it's possible to find it.
GMAT's explanation makes sense, but I think mine is better.

#### ironsheep

##### Junior Member
So if we add 4 to n = 7q + 3 and n = 5p + 1, they become n = 7q +7 and n = 5p +5, doesn't matter what variables you use to substitute for q and p the answers will always be divisible by 5 or 7. For example, 5 times 30 plus 4 equals 154 and that is divisible by 7, so it is a multiple of 35 and that is all this question is asking??

#### lev888

##### Full Member
So if we add 4 to n = 7q + 3 and n = 5p + 1, they become n = 7q +7 and n = 5p +5, doesn't matter what variables you use to substitute for q and p the answers will always be divisible by 5 or 7. For example, 5 times 30 plus 4 equals 154 and that is divisible by 7, so it is a multiple of 35 and that is all this question is asking??
No, you can't say "doesn't matter what variables you use to substitute for q and p".
7q+7 and 5p+5 are divisible by 5 and 7 for those p and q that satisfy 7q+7 = 5p+5. You don't need to know what they are to find k. But you can't use any p and q to make 7q+7 and 5p+5 divisible by both 5 and 7.

#### ironsheep

##### Junior Member
Why would you need to find the variables for p and q when the answer is 4 because adding four makes everything divisible for 7 and 5? 4 is the answer, you don't need to go further.

#### lev888

##### Full Member
Why would you need to find the variables for p and q when the answer is 4 because adding four makes everything divisible for 7 and 5? 4 is the answer, you don't need to go further.
Yes, that's why I wrote "You don't need to know what they are to find k".
But you statement "doesn't matter what variables you use to substitute for q and p the answers will always be divisible by 5 or 7" is incorrect. Do you see why only certain pairs of p and q work?
It's one thing to say "we don't need to find p and q to find k". It's quite different to say "it doesn't matter what q and p are, the answers will always be divisible by 5 or 7".

#### ironsheep

##### Junior Member
Well, no matter what postive integer you use to subsitite the q and p, the answer will always be divisble by 5 or 7. If p was 100, 310, 34, or 39 etc... or if q was 300, 222, 65, 56, etc... the answers would still be divisble by 5 or 7.

#### lev888

##### Full Member
Well, no matter what postive integer you use to subsitite the q and p, the answer will always be divisble by 5 or 7. If p was 100, 310, 34, or 39 etc... or if q was 300, 222, 65, 56, etc... the answers would still be divisble by 5 or 7.
By 5 OR 7. Yes. But in order to be divisible by 35 the number has to be divisible by 5 AND 7.
q = 1, 7q+7 = 14 - divisible by 7 but not by 5.

#### ironsheep

##### Junior Member
Thank you for the replies and help