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16 Sep 2014, 01:01
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If \(x\) and \(y\) are positive integers, what is the remainder when \(xy\) is divided by 4?
(1) When \(x\) is divided by 4, the remainder is 3
(2) When \(y\) is divided by 4, the remainder is 2
(1) When \(x\) is divided by 4, the remainder is 3
(2) When \(y\) is divided by 4, the remainder is 2
16 Sep 2014, 01:01
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Official Solution:
If \(x\) and \(y\) are positive integers, what is the remainder when \(xy\) is divided by 4?
Note: A positive integer \(a\) divided by a positive integer \(d\) that yields a remainder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called the quotient and \(r\) is called the remainder; note here that \(0 \le r \lt d\) (the remainder is a non-negative integer and always less than the divisor).
(1) When \(x\) is divided by 4, the remainder is 3.
Not sufficient as no info about \(y\).
(2) When \(y\) is divided by 4, the remainder is 2.
Not sufficient as no info about \(x\).
(1)+(2) From (1) we have \(x=4q+3\) and from (2) \(y=4p+2\), so \(xy=(4q+3)(4p+2)=16qp+8q+12p+6\). All terms except the last are divisible by 4, and the last term, 6, yields a remainder of 2 when divided by 4. Sufficient
Another approach: Possible values for \(x\) include: 3, 7, 11, ... and for \(y\): 2, 6, 10, 14, ... By trying several of these values, we see that \(xy\) always yields a remainder of 2 when divided by 4.
Answer: C
If \(x\) and \(y\) are positive integers, what is the remainder when \(xy\) is divided by 4?
Note: A positive integer \(a\) divided by a positive integer \(d\) that yields a remainder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called the quotient and \(r\) is called the remainder; note here that \(0 \le r \lt d\) (the remainder is a non-negative integer and always less than the divisor).
(1) When \(x\) is divided by 4, the remainder is 3.
Not sufficient as no info about \(y\).
(2) When \(y\) is divided by 4, the remainder is 2.
Not sufficient as no info about \(x\).
(1)+(2) From (1) we have \(x=4q+3\) and from (2) \(y=4p+2\), so \(xy=(4q+3)(4p+2)=16qp+8q+12p+6\). All terms except the last are divisible by 4, and the last term, 6, yields a remainder of 2 when divided by 4. Sufficient
Another approach: Possible values for \(x\) include: 3, 7, 11, ... and for \(y\): 2, 6, 10, 14, ... By trying several of these values, we see that \(xy\) always yields a remainder of 2 when divided by 4.
Answer: C
07 Oct 2023, 12:45
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
