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by alex.gellatly » Wed Jul 11, 2012 11:36 pm
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?

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by das.ashmita » Thu Jul 12, 2012 1:08 am
is it 35?

I am not sure about my approach.

x and y both can be represented in terms of 5p+3 and 7q+4
Therefore, 5p+3 = 7q+4 => 5p-7q = 1

By picking nos satisfying the above equation, I got x and y as 53 and 18 respectively.

x-y = 53-18 = 35.
Since no other option is a factor of 35, hence ans is 35
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by GMATGuruNY » Thu Jul 12, 2012 2:42 am
alex.gellatly wrote: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?

12
15
20
28
35
When positive integer x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3 = 3, 8, 13, 18, 23...

When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4 = 4, 11, 18...

The smallest value common to both lists is 18.
This means that x is 18 more than a multiple of the product of the two divisors (5 and 7):
x = 35c + 18 = 18, 53, 88, 123...

Since all of the same conditions apply to y:
y = 35d + 18 = 18, 53, 88, 123...

Thus:
x-y = (35c + 18) - (35d + 18) = 35(c-d), implying that x-y is a multiple of 35.

The correct answer is E.
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by alex.gellatly » Thu Jul 12, 2012 6:15 pm
GMATGuruNY wrote:
alex.gellatly wrote: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?

12
15
20
28
35
When positive integer x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3 = 3, 8, 13, 18, 23...

When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4 = 4, 11, 18...

The smallest value common to both lists is 18. I get confused about here
This means that x is 18 more than a multiple of the product of the two divisors (5 and 7):
x = 35c + 18 = 18, 53, 88, 123...

Since all of the same conditions apply to y:
y = 35d + 18 = 18, 53, 88, 123...

Thus:
x-y = (35c + 18) - (35d + 18) = 35(c-d), implying that x-y is a multiple of 35.

The correct answer is E.
Thank you very much... But I am still very confused.
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by GMATGuruNY » Fri Jul 13, 2012 7:28 am
alex.gellatly wrote:Thank you very much... But I am still very confused.
When x is divided by 5, the remainder is 3.
When x is divided by 7, the remainder is 4.

These statements imply that when x is divided by both 5 and 7 -- in other words, when x is divided by 35 -- there will be a constant remainder R.
Put another way, x is R more than a multiple of 35:
x = 35c + R.

To determine the value of R:
Make a list of values that satisfy the first statement:
When x is divided by 5, the remainder is 3.
x = 5a + 3 = 3, 8, 13, 18...
Make a list of values that satisfy the second statement:
When x is divided by 7, the remainder is 4.
x = 7b + 4 = 4, 11, 18...
The value of R is the SMALLEST VALUE COMMON TO BOTH LISTS:
R = 18.

Putting it all together:
x = 35c + 18.

Another example:
When x is divided by 3, the remainder is 1.
x = 3a + 1 = 1, 4, 7, 10, 13...
When x is divided by 11, the remainder is 2.
x = 11b + 2 = 2, 13...

Thus, when x is divided by both 3 and 11 -- in other words, when x is divided by 33 -- the remainder will be 13 (the smallest value common to both lists).
x = 33c + 13 = 13, 46, 79...
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