Math-Book-GMAT-Club

Example #10 (hard)
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
When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively:
3, 8, 13, 18, 23, ...) and (x could be 4, 11, 18, 25, ...).
(x could be
We can derive general formula based on above two statements the same way as for the example above:
Divisor will be the least common multiple of above two divisors 5 and 7, hence 35.
Remainder will be the first common integer in above two patterns, hence 18. So, to satisfy both this conditions x
must be of a type (18, 53, 88, ...);
The same for y (as the same info is given about y): ;
Answer: E. Discuss this question HERE.
Example #11 (hard)
. Thus must be a multiple of 35.
If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by 4?
(1) When p is divided by 8, the remainder is 5
(2) x – y = 3
(1) When p is divided by 8, the remainder is 5. This implies that . Since given
that , then ‐‐
> .
So,
. Now,
if then and
if then , so in any
case ‐‐> ‐‐> in order to be multiple of 4 must be multiple of
16 but as we see it's not, so is not multiple of 4. Sufficient.
(2) x – y = 3 ‐‐> ‐‐> but not sufficient to say whether it's multiple of 4.
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