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10 Nov 2021, 09:00
Kudos
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
75% (02:08) correct
25%
(02:31)
wrong
based on 155
sessions
History
Date
Time
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Not Attempted Yet
The integers \(x\) and \(y\) are positive, \(x > y + 8\), and \(y > 8\). What is the remainder when \(x^2 − y^2\) is divided by 8?
(1) The remainder when \(x + y\) is divided by 8 is 7.
(2) The remainder when \(x – y\) is divided by 8 is 5.
(1) The remainder when \(x + y\) is divided by 8 is 7.
(2) The remainder when \(x – y\) is divided by 8 is 5.
11 Nov 2021, 04:12
Kudos
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Bunuel
Concept: Remainder get multiplied and added
\(x > y + 8\), and \(y > 8\) MEANS \(y\geq 9\) and \(x\geq 18\)
\(x^2-y^2 = (x-y)(x+y)\). So we have to look for remainder when the sum and difference of x and y are divided by 8.
(1) The remainder when \(x + y\) is divided by 8 is 7.
Nothing about x-y.
Say x=20 and y = 11, the remainder of \((20-11)(20+11) = 9*31=(1+8)(24+7)\) is 1*7.
Say x=21 and y = 10, the remainder of \((21-10)(21+10) = 11*31=(3+8)(24+7)\) is 3*7 or 21 or 16+5 or 5 is the remainder
Insuff
(2) The remainder when \(x – y\) is divided by 8 is 5.
Nothing about x+y.
Say x=23 and y = 10, the remainder of \((23-10)(23+10) = 13*33=(5+8)(32+1)\) is 5*1.
Say x=22 and y = 9, the remainder of \((22-9)(22+9) = 13*31=(5+8)(24+7)\) is 5*7 or 35 or 32+3 or 3 is the remainder
Insuff
Combined
(x+y)(x-y) will give the remainder as the product of remainders of x+y and x-y separately.
So remainder = 7*5 = 35 =32+3.
Thus, 3 is the answer.
Sufficient
C
12 Nov 2021, 10:54
Kudos
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Bunuel
Breaking Down the Info:
We are interested in the expression \((x + y)(x - y)\), hence \(x + y\) and \(x - y\) respectively. We are also given \(x - y > 8\) and \(y > 8\); it follows that \(x > 16\).
Statement 1 Alone:
With the same values of \(x + y\), we can have different values of \(x - y\), which will result in different values of \(x^2 - y^2\) and different remainders (we can increase or decrease \(x - y\) in increments of 2 for the same x + y values). So this is insufficient.
Statement 2 Alone:
Using the same logic in statement 1, this is insufficient. (We may increase/decrease x + y in increments of 2 while holding x - y the same).
Both Statements Combined:
Write \(x + y\) as \(8i + 7\) and \(x - y\) as \(8j + 5\). Then if we multiply both we have \(64i*j + 40i + 56j + 35\). All the terms are a multiple of 8 except 35, so diving this by 8 will give us the same remainder as \(35/8 = 4 + 3/8\), a remainder of 3. Then combined it is sufficient.
Answer: C
