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03 Aug 2023, 01:30
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Difficulty:
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Question Stats:
65% (02:10) correct
35%
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wrong
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If \(a\) and \(b\) are positive integers and \(a > b\), what is the remainder when \(a^2 - 2ab + b^2\) is divided by \(9\)?
(1) The remainder when \(a-b\) is divided by \(3\) is \(2\).
(2) The remainder when \(a-b\) is divided by \(9\) is \(2\).
(1) The remainder when \(a-b\) is divided by \(3\) is \(2\).
(2) The remainder when \(a-b\) is divided by \(9\) is \(2\).
03 Aug 2023, 08:23
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Given that a and b are positive integers and \(a > b\) and we need to find what is the remainder when \(a^2 - 2ab + b^2\) is divided by \(9\)?
\(a^2 - 2ab + b^2\) = \((a-b)^2\)
STAT 1: The remainder when \(a-b\) is divided by \(3\) is \(2\).
Dividend = Divisor * Quotient + Remainder
=> a-b = 3*k + 2 (where k is a quotient and is a non-negative integer) = 3k + 2
=> \((a-b)^2\) = \((3k + 2)^2\) = \(9k^2 + 2*3k*2 + 2^2\) = \(9k^2\) + 12k + 4
=> Remainder when \((a-b)^2\) is divided by 9 = Remainder when \(9k^2\) + 12k + 4 is divided by 9 = Remainder when \(9k^2\) is divided by 9 + Remainder when 12k + 4 is divided by 9 = 0 + Remainder when 12k + 4 is divided by 9 = Remainder when 12k + 4 is divided by 9
Now depending of value of k we can get many values of remainders
k = 1, Remainder will be Remainder when 12*1 + 4 is divided by 9 = Remainder when 16 is divided by 9 = 7
k = 2, Remainder will be Remainder when 12*2 + 4 is divided by 9 = Remainder when 28 is divided by 9 = 1
=> We can get multiple values of remainders
=> NOT SUFFICIENT
STAT 2: The remainder when \(a-b\) is divided by \(9\) is \(2\).
=> a-b = 9*k + 2 (where k is a quotient and is a non-negative integer) = 9k + 2
=> \((a-b)^2\) = \((9k + 2)^2\) = \((9k)^2 + 2*9k*2 + 2^2\) = \(9*(9k^2 + 4k)\) + 4
=> Remainder when \((a-b)^2\) is divided by 9 = Remainder when \(9*(9k^2 + 4k)\) + 4 is divided by 9 = Remainder when \(9*(9k^2 + 4k)\) is divided by 9 + Remainder when 4 is divided by 9 = 0 + 4 = 4
=> SUFFICIENT
So, Answer will be B
Hope it helps!
Watch the following video to learn the Basics of Remainders
\(a^2 - 2ab + b^2\) = \((a-b)^2\)
STAT 1: The remainder when \(a-b\) is divided by \(3\) is \(2\).
Dividend = Divisor * Quotient + Remainder
=> a-b = 3*k + 2 (where k is a quotient and is a non-negative integer) = 3k + 2
=> \((a-b)^2\) = \((3k + 2)^2\) = \(9k^2 + 2*3k*2 + 2^2\) = \(9k^2\) + 12k + 4
=> Remainder when \((a-b)^2\) is divided by 9 = Remainder when \(9k^2\) + 12k + 4 is divided by 9 = Remainder when \(9k^2\) is divided by 9 + Remainder when 12k + 4 is divided by 9 = 0 + Remainder when 12k + 4 is divided by 9 = Remainder when 12k + 4 is divided by 9
Now depending of value of k we can get many values of remainders
k = 1, Remainder will be Remainder when 12*1 + 4 is divided by 9 = Remainder when 16 is divided by 9 = 7
k = 2, Remainder will be Remainder when 12*2 + 4 is divided by 9 = Remainder when 28 is divided by 9 = 1
=> We can get multiple values of remainders
=> NOT SUFFICIENT
STAT 2: The remainder when \(a-b\) is divided by \(9\) is \(2\).
=> a-b = 9*k + 2 (where k is a quotient and is a non-negative integer) = 9k + 2
=> \((a-b)^2\) = \((9k + 2)^2\) = \((9k)^2 + 2*9k*2 + 2^2\) = \(9*(9k^2 + 4k)\) + 4
=> Remainder when \((a-b)^2\) is divided by 9 = Remainder when \(9*(9k^2 + 4k)\) + 4 is divided by 9 = Remainder when \(9*(9k^2 + 4k)\) is divided by 9 + Remainder when 4 is divided by 9 = 0 + 4 = 4
=> SUFFICIENT
So, Answer will be B
Hope it helps!
Watch the following video to learn the Basics of Remainders
03 Aug 2023, 16:38
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E=a²-2ab+b² = (a+b)² = 9k+y, where k,y are integers and 0<=y<9. Lets analyze what statement is enough to determine y:
1) a-b=3m+2 => (a-b)²=9(m²)+(3m+4). Here, y = 3m+4 (mod9) can be 4 or 7. INSUFF
2) a-b=9p+2 => (a-b)²=9(9p²+4p)+4. So y = 4. SUFF.
Answer: B)
1) a-b=3m+2 => (a-b)²=9(m²)+(3m+4). Here, y = 3m+4 (mod9) can be 4 or 7. INSUFF
2) a-b=9p+2 => (a-b)²=9(9p²+4p)+4. So y = 4. SUFF.
Answer: B)
