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17 Dec 2022, 08:35
Kudos
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C
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Difficulty:
75%
(hard)
Question Stats:
44% (02:08) correct
56%
(02:06)
wrong
based on 41
sessions
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If x and y are positive integers, is the sum x + y divisible by 4 ?
(1) When the sum \(23^x + 25^y\) is divided by 10, the remainder is 8.
(2) When \(22^y\) is divided by 10, the remainder is 8.
(1) When the sum \(23^x + 25^y\) is divided by 10, the remainder is 8.
(2) When \(22^y\) is divided by 10, the remainder is 8.
17 Dec 2022, 08:53
Kudos
Bookmarks
If x and y are positive integers, is the sum x + y divisible by 4 ?
(1) When the sum \(23^x + 25^y\) is divided by 10, the remainder is 8.
Not sufficient.
(2) When \(22^y\) is divided by 10, the remainder is 8.
(1)+(2) From above: x + y = (4m + 1) + (4n + 3) = 4(m + n + 1). Thus, the remainder when x + y is divided by 4 is 0. Sufficient.
Answer: C.
(1) When the sum \(23^x + 25^y\) is divided by 10, the remainder is 8.
- The above means that the units digit of \(23^x + 25^y\) is 8.
The units digit of 25^y, when y is a positive integer, is always 5 (so we cannot say anything about y from this statement).
The above means that the units digit of 23^x must be 3 (so that the units digit of \(23^x + 25^y\) is 8). Thus x must be 1, 5, 9, ... (basically 4m + 1, for nonnegative integer m)
Not sufficient.
(2) When \(22^y\) is divided by 10, the remainder is 8.
- The above means that the units digit of \(23^y\) is 8. Thus, y is 3, 7, 11, ... (basically 4n + 3, for nonnegative integer n).
(1)+(2) From above: x + y = (4m + 1) + (4n + 3) = 4(m + n + 1). Thus, the remainder when x + y is divided by 4 is 0. Sufficient.
Answer: C.
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