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A_n=2^n-1 for all integers n >=1
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10 Jan 2023, 11:24
10 Jan 2023, 11:24
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\(A_n=2^n-1\) for all integers \(n \geq 1\)
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Quantity A |
Quantity B |
The unit digit of \(A_{26}\) |
The unit digit of \(A_{34}\) |
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
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A_n=2^n-1 for all integers n >=1
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14 Jan 2023, 22:33
14 Jan 2023, 22:33
1
Given that \(A_n=2^n-1\) for all integers \(n \geq 1\)
Quantity A
The unit digit of \(A_{26}\) = Unit's digit of (\(2^{26} - 1\))
Unit's digit of \(2^1\) = 2
Unit's digit of \(2^2\) = 4
Unit's digit of \(2^3\) = 8
Unit's digit of \(2^4\) = 6
Unit's digit of \(2^5\) = 2
So, unit's digit of power of 2 repeats after every \(4^{th}\) number.
=> We need to divided 26 by 4 and check what is the remainder
=> 26 divided by 4 gives 2 remainder
=> \(2^{26}\) will have the same unit's digit as \(2^2\) = 4
=> Unit's digits of \(2^{26}\) = 4
=> Unit's digit of \(2^{26} - 1\) = 4 - 1 = 3
Quantity B
The unit digit of \(A_{34}\) = Unit's digit of (\(2^{34} - 1\))
34 divided by 4 gives 2 remainder
=> \(2^{34}\) will have the same unit's digit as \(2^2\) = 4
=> Unit's digits of \(2^{34}\) = 4
=> Unit's digit of \(2^{34} - 1\) = 4 - 1 = 3
Clearly, Quantity A(3) = Quantity B(3)
So, Answer will be C
Hope it helps!
Watch the following video to learn How to find Unit's digit
Quantity A
The unit digit of \(A_{26}\) = Unit's digit of (\(2^{26} - 1\))
Unit's digit of \(2^1\) = 2
Unit's digit of \(2^2\) = 4
Unit's digit of \(2^3\) = 8
Unit's digit of \(2^4\) = 6
Unit's digit of \(2^5\) = 2
So, unit's digit of power of 2 repeats after every \(4^{th}\) number.
=> We need to divided 26 by 4 and check what is the remainder
=> 26 divided by 4 gives 2 remainder
=> \(2^{26}\) will have the same unit's digit as \(2^2\) = 4
=> Unit's digits of \(2^{26}\) = 4
=> Unit's digit of \(2^{26} - 1\) = 4 - 1 = 3
Quantity B
The unit digit of \(A_{34}\) = Unit's digit of (\(2^{34} - 1\))
34 divided by 4 gives 2 remainder
=> \(2^{34}\) will have the same unit's digit as \(2^2\) = 4
=> Unit's digits of \(2^{34}\) = 4
=> Unit's digit of \(2^{34} - 1\) = 4 - 1 = 3
Clearly, Quantity A(3) = Quantity B(3)
So, Answer will be C
Hope it helps!
Watch the following video to learn How to find Unit's digit
gmatclubot
A_n=2^n-1 for all integers n >=1 [#permalink]
14 Jan 2023, 22:33
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