When positive integer x is divided by 5, the remainder is 3

Question

When positive integer x is divided by 5, the remainder is 3
i.e. x = 5a+3
i.e. x may be {3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58,...}

when x is divided by 7, the remainder is 4
i.e. x = 7b+4
i.e. x may be {4, 11, 18, 25, 32, 39, 46, 53, 60, 67,...}

Looking at common numbers we see that
x may be {18, 53, 88...}

A. 12
B. 15
C. 20
D. 28
E. 35

Bunuel · Accepted Answer

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?

A. 12
B. 15
C. 20
D. 28
E. 35

Here's my solution:

Concept: Whenever you are subtracting two numbers, you are completely subtracting the remainders..... let's call this (I)
Let me explain: Consider X = 21 & Y = 11 -------- (X mod 5) = 1 || (Y mod 5) = 1 - hence the remainder is 1 for both

Now, from (I),
Subtracting both X and Y should remove the remainder.

X-Y = 21 - 11 = 10 ----> (10 mod 5) = 0, hence (I) stands.

Let's project this concept into our numerical,

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4 -
What if there were no remainders, X would have been a multiple of 5 and 7, hence -----> a multiple of 5 *7 = 35

When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4.
What if there were no remainders, Y would have been a multiple of 5 and 7, hence -----> a multiple of 5 *7 = 35

If x > y, which of the following must be a factor of x - y?
Now since the question itself is asking you to subtract both the numbers (X-Y) remainders will Nil out, therefore the X-Y has to be nothing but a multiple of 35.

Hence the Answer is (E)

lchen · Answer

I do the "for dummies" way on this, because it is the first thing that popped in my mind, and the first thing I would have tried. It actually doesn't take that long on paper. Easily under 2 minutes.

5x+3: 
5+3=8 
10+3=13 
15+3=   18    
20+3=23 
25+3=28 
30+3=33 
35+3=38 
40+3=43 
45+3=48 
50+3=   53

7x+4:
7+4=11
14+4=   18    
21+4=25
28+4=32
35+4=39
42+4=46
49+4=   53    
56+4=60
63+4=67
70+4=74

The thing about PS problems is that there can only be one answer. So as long as you can find it one time, it will be the same for all other times.
The question is asking for the larger number minus the smaller number
so, 53-18 = 35

Answer is E

ScottTargetTestPrep · Answer

Let’s list some integers that have a remainder of 3 when divided by 5:

3, 8, 13, 18, 23, …

Let’s list some integers that have a remainder of 4 when divided by 7:

4, 11, 18, 25, …

We see that 18 is common in both lists and is the smallest positive integer that satisfies both conditions. Thus, y could be 18. To get a value for x, we can simply add the LCM of 5 and 7, i.e, 35, to 18. So, x could be 53 (notice that 53 also satisfies both conditions). We see that in this case, x - y = 35, and of all the choices, only 35 is factor of x - y.

Alternate Solution:

Since x produces a remainder of 3 when divided by 5, we can write x = 5p + 3 for some integer p.

Since x produces a remainder of 4 when divided by 7, we can write x = 7q + 4 for some integer q.

Since y produces a remainder of 3 when divided by 5, we can write y = 5s + 3 for some integer s.

Since y produces a remainder of 4 when divided by 7, we can write y = 7r + 4 for some integer r.

If we subtract the third equality from the first, we obtain: x - y = 5p - 5s = 5(p - s). Thus, x - y is a multiple of 5.

If we subtract the fourth equality from the second, we obtain: x - y = 7q - 7r = 7(q - r). Thus, x - y is a multiple of 7.

Since x - y is a multiple of both 5 and 7, it is a multiple of the LCM of 5 and 7 as well, namely 35.

Answer: E

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on remainders problems: remainders-144665.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199

AccipiterQ · Answer

there must be some other way to solve this; I have no idea how anyone that reads that question could sit there and think of what you wrote out, in under two minutes. It's a great solution, but I think there must be some other way to crack this nut.

Bunuel · Answer

If you know the trick to derive general formula you can solve the question just under 2 minutes.

Else, you can find common numbers (18 and 53) and see that 53-18 is a multiple of only 35 from answer choices.

AccipiterQ · Answer

Actually, on this problem, if 5 is a factor of it, as is 7, couldn't you just look at the stem, and multiply 5 by 7 without doing any extra work, to get the answer?

Bunuel · Answer

No. Neither 5 nor 7 is a factor of either x or y.

KarishmaB · Answer

Yes, there is. You just read the question and the answer will be there by the time you are done (doing everything Bunuel did above).

Consider the question one line at a time:

"When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4." 
Think - You will get x by finding the first such value which when divided by 5 leaves remainder 3 and when divided by 7 leaves remainder 4.
x will be 35N + First such value

"When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4." 
Go back and compare the divisors and remainders. You see that they are exactly the same. This means y also belongs to the same series.
y = 35M + First such value

So you know now that x will be some multiple of 35 plus a number and y will be another multiple of 35 plus the same number. So x and y will differ by some multiple of 35.

"If x > y, which of the following must be a factor of x - y?" 
So 35 must be a factor of x-y.

Answer (E)

ajdse22 · Answer

There is one other way to solve this problem. I earlier went with enumerating choices and it was taking time. So here is another way
x>y is given
x=5a1+3 and y=5a2+3..
also x=7b1+4 and y = 7b2+4
x-y = 5a1-5a2+3+3=> 5(a1-a2)
also x-y => 7b1-7b2+4-4=> 7(b1-b2).
so x-y is a multiple of both 5 and 7. 35 fits the bill it is a factor of 7 and 5.

gracie · Answer

because the divisors and remainders are the same for both dividends,
all values of x-y will equal a multiple of the product of the two divisors, 5 and 7
5*7=35
E

DanTheGMATMan · Answer

Classic remainders deal:

https://www.youtube.com/watch?v=cruMFvn3xf4

fayey · Answer

Hi Bunuel, 
why we can know  x=35m+18  according  x=5q+3 & x=7p+4? 
 Is the conclusion drawn by exhaustive method?

Bunuel · Answer

Please check the highlighted parts.

Hope it helps.

fayey · Answer

Thank you! 
I just confused about the common remainder. So anyway we have to list them separately and find the first common integer in the above two patterns to be the new remainder.

aryanrgupta · Answer

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y? 
 
A. 12
B. 15
C. 20
D. 28
E. 35

Here's my solution:

Concept: Whenever you are subtracting two numbers, you are completely subtracting the remainders..... let's call this  (I)  
Let me explain: Consider X = 21 & Y = 11 -------- (X mod 5) = 1 || (Y mod 5) = 1 - hence the remainder is 1 for both

Now, from (I),
 Subtracting both X and Y should remove the remainder.

X-Y = 21 - 11 = 10 ----> (10 mod 5) = 0, hence (I) stands.

Let's project this concept into our numerical,

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4 - 
 What if there were no remainders, X would have been a multiple of 5 and 7, hence -----> a multiple of 5 *7 = 35

When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. 
 What if there were no remainders, Y would have been a multiple of 5 and 7, hence -----> a multiple of 5 *7 = 35

If x > y, which of the following must be a factor of x - y?  
Now since the question itself is asking you to subtract both the numbers (X-Y) remainders will Nil out, therefore the X-Y has to be nothing but a multiple of 35.

Hence the Answer is (E)

GMATinsight · Answer

There are two methods to solving this question but the simplest one is here

Let, Positive Integer = N

When positive integer x is divided by 5, the remainder is 3
i.e. x = 5a+3
i.e. x may be {3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58,...}

when x is divided by 7, the remainder is 4
i.e. x = 7b+4
i.e. x may be {4, 11, 18, 25, 32, 39, 46, 53, 60, 67,...}

Looking at common numbers we see that
x may be {18, 53, 88...}

Positive integer y is divided by 5, the remainder is 3;
i.e. y = 5c+3
i.e. y may be {3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58,...}

and when y is divided by 7, the remainder is 4
i.e. x = 7b+4
i.e. x may be {4, 11, 18, 25, 32, 39, 46, 53, 60, 67,...}

Looking at common numbers we see that
y may be {18, 53, 88...}

we know that x > y

so, we can choose x to be 53 when y is 18
so x-y = 53-18 = 35
Answer: Option E

To check more videos on Number properties check the kink Below (200 Videos)
 Number Properties Videos

ManifestDreamMBA · Answer

x = 5a+3, 3,8,13, 18 ,23,28,33, ....
x = 7b+4, 4,11, 18 ,25,32,....

Thinking of generic formula, x = 35k+18

y follows the same generic formula, y = 35p+18

x - y = 35(k-p)

Answer: 35

egmat · Answer

Let me help you see the elegant pattern behind this remainder question.

When I first encounter students struggling with this type of problem, they often try to find the exact values of x and y using the Chinese Remainder Theorem. That's unnecessarily complex for the GMAT. Let me show you the streamlined approach.

Key Insight:  Both x and y have  identical  remainder patterns:
- Remainder 3 when divided by 5
- Remainder 4 when divided by 7

Step 1: Express the conditions algebraically 
For any number with these remainder properties, we can write:
$x = 5k + 3$ for some integer k
$x = 7m + 4$ for some integer m

Similarly for y:
$y = 5j + 3$ for some integer j
$y = 7n + 4$ for some integer n

Step 2: Apply the difference property 
Here's the crucial pattern - when two numbers have identical remainders, their difference eliminates those remainders:

$x - y = (5k + 3) - (5j + 3) = 5(k - j)$
$x - y = (7m + 4) - (7n + 4) = 7(m - n)$

This tells us that $x - y$ is divisible by both 5 and 7.

Step 3: Find what MUST divide x - y 
Since 5 and 7 are coprime (share no common factors), and $x - y$ is divisible by both, $x - y$ must be divisible  by their product:  5*7 = 35

Want to master this remainder pattern that appears in 3-4 GMAT questions yearly?  The complete solution  walks through the coprime divisor principle and shows you how to spot this pattern instantly.

The framework there also covers what happens when divisors aren't coprime - a common trap the GMAT loves to test.

When positive integer x is divided by 5, the remainder is 3

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GMAT Problem Solving (PS) Questions

When positive integer x is divided by 5, the remainder is 3

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615 to 715 in 15 Days (Ria's Strategies)

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