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Difficulty:
35%
(medium)
Question Stats:
65% (01:14) correct
35%
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wrong
based on 128
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If x and y are positive integers, what is the units digit of \(3^{4x+2y+6}\)?
(1) x = 1
(2) y = 2
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GMAT 1: 720 Q49 V40
Posts: 939
Units digit of 3^(4x+2y+6)
(1) x = 1
3^(10+2y)=9^(5+y)
If y is odd, the units digit is 1. On the other hand, If y is even, the units digit is 9.
NOT SUFFICIENT
(2) y = 2
3^(10+4x)=9^(5+2x)
(5+2x) is always odd, thus the resulting units digit is 9.
SUFFICIENT
FINAL ANSWER IS (B)
Posted from my mobile device
(1) x = 1
3^(10+2y)=9^(5+y)
If y is odd, the units digit is 1. On the other hand, If y is even, the units digit is 9.
NOT SUFFICIENT
(2) y = 2
3^(10+4x)=9^(5+2x)
(5+2x) is always odd, thus the resulting units digit is 9.
SUFFICIENT
FINAL ANSWER IS (B)
Posted from my mobile device
If x and y are positive integers, what is the units digit of \(3^{4x+2y+6}\)?
\(3^{4x}*3^{2y}*3^6\)
Unit place of \(3^{4x}\) will always be 1 as 3's cyclicity of 4(3^4 = 81)
Unit place of \(3^{2y}\) will be 9 when y is odd and 1 when y is even
Unit place of \(3^6\) will always be 9
Thus, unit place of \(3^{4x+2y+6}\) is either 1(1*9*9 = 81) OR 9(1*1*9=9)
Hence only 'y' governs the value of \(3^{4x+2y+6}\)
(1) x = 1
y is not defined.
INSUFFICIENT.
(2) y = 2
Unit place of \(3^{4x+2y+6}\) is 1.
SUFFICIENT.
Answer B.
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\(3^{4x}*3^{2y}*3^6\)
Unit place of \(3^{4x}\) will always be 1 as 3's cyclicity of 4(3^4 = 81)
Unit place of \(3^{2y}\) will be 9 when y is odd and 1 when y is even
Unit place of \(3^6\) will always be 9
Thus, unit place of \(3^{4x+2y+6}\) is either 1(1*9*9 = 81) OR 9(1*1*9=9)
Hence only 'y' governs the value of \(3^{4x+2y+6}\)
(1) x = 1
y is not defined.
INSUFFICIENT.
(2) y = 2
Unit place of \(3^{4x+2y+6}\) is 1.
SUFFICIENT.
Answer B.
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madgmat2019
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what ever the value of x its not effecting the exponential value...and the value is solely dependent on the value of y if its odd or even...
so 1 is insufficient
2 is sufficient
OA:B
so 1 is insufficient
2 is sufficient
OA:B
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#1
x=1
3^2y+10
the unit digit will vary with each value of y = 1,2 insufficient
#2
y=2
3^4x+10
the unit digit shall be 9 always
IMO B sufficient
If x and y are positive integers, what is the units digit of 3^4x+2y+6?
(1) x = 1
(2) y = 2
x=1
3^2y+10
the unit digit will vary with each value of y = 1,2 insufficient
#2
y=2
3^4x+10
the unit digit shall be 9 always
IMO B sufficient
If x and y are positive integers, what is the units digit of 3^4x+2y+6?
(1) x = 1
(2) y = 2
We are to determine the unit digit of 3^(4x+2y+6) given that x and y are positive integers.
3^(4x+2y+6) = 3^4x * 3^2y * 3^6
We know that the cyclicity of 3 is 4. Hence the value of x is irrelevant since it will always lead to the same unit digit. In addition, 3^6 has the same unit digit as 3^2 = 9. 3^2y, on the other hand, is not fixed. when y is odd, then 3^2y has the same cyclicity as 3^2. When y is even however, 3^2y has the same cyclicity of 3^4, hence we need the value of y in order to determine the unit digit of 3^(4x+2y+6).
Statement 1: x=1
This is insufficient. We need only the value of y since the value of x is immaterial to the determination of the unit digit of 3.
Statement 2: y=2
This is sufficient since we have been provided the only missing piece of the puzzle, which is the value of y. We now know that 3^(4x+2y+6) has the same unit digits as 3^2 = 9.
The answer is therefore B.
3^(4x+2y+6) = 3^4x * 3^2y * 3^6
We know that the cyclicity of 3 is 4. Hence the value of x is irrelevant since it will always lead to the same unit digit. In addition, 3^6 has the same unit digit as 3^2 = 9. 3^2y, on the other hand, is not fixed. when y is odd, then 3^2y has the same cyclicity as 3^2. When y is even however, 3^2y has the same cyclicity of 3^4, hence we need the value of y in order to determine the unit digit of 3^(4x+2y+6).
Statement 1: x=1
This is insufficient. We need only the value of y since the value of x is immaterial to the determination of the unit digit of 3.
Statement 2: y=2
This is sufficient since we have been provided the only missing piece of the puzzle, which is the value of y. We now know that 3^(4x+2y+6) has the same unit digits as 3^2 = 9.
The answer is therefore B.
Quote:
Cycles of the powers of THREE: [3,9,7,1] = 4
We need to find the remainder of (4x+2y+6)/4 to find the units digits
(1) x = 1 insufic
(4x+2y+6)=4+2y+6=2y+10
2y+10/4: remainder 0 or 2
y=1:12/4=remainder0
y=2:14/4=remainder2
(2) y = 2 sufic
(4x+2y+6)=4x+4+6=4x+10
(4x+10)/4: 10/4 remainder2
Ans (B)
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Quote:
Statement (1)
\(x=1 => 3^{2y+10}\)
=>Not suff
Statement (2)
\(y=2 => 3^{4x+10}\)
=> Not suff
Combine two statements: => unit's digit of \(3^{14}\) is 9
=> Choice C
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