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If a and b are positive integers than x=4^a & y=9^b [#permalink]
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10 Oct 2011, 10:22
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If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy? 1 4 5 7 8 The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance
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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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10 Oct 2011, 11:22
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Answer is 4 :
Cycle for 4^a is 4 or 6 : 4 16 64 256 1024 4096 cycle for 9^b is 1 or 9 : 9 81 729 6561
so possible units digit of xy = 4 or 6
Last edited by MisterQ on 10 Oct 2011, 11:27, edited 1 time in total.



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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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10 Oct 2011, 11:27
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shrive555 wrote: If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy? 1 4 5 7 8 The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance x=4^a a is +ve integer. a=1; x= 4a=2; x=4^2=1 6a=3; x=4^3=6 4a^4=4*4= 6You see; the units digit of 4^x is either 4 OR 6. Likewise; units digit of 9^x is either 9 OR 1. Possible units digit; 4*9=3 64*1= 46*9=5 46*1= 66 is not in the options, but 4 is. Ans: "B"
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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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10 Oct 2011, 14:50
got it.. ..Thanks guys !!
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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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10 Oct 2011, 20:50
x=4^a y=9^b
xy =(4^a)*(9^b)
when a=1,b=1 , xy 's unit digit =i.e unit's digit of (4*9= 36) i.e 6 a=1,b=2, xy 's unit digit = i.e units digit of (4*81) i.e 4
Answer is B



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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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11 Oct 2011, 05:01
Cyclisity of 4^n is 4,6,4,6 9^n is 9,1,9,1
given the question, we can have
9*4=36, 6 9*6=54, 4 1*4=4, 4 1*6=6, 6
therefore 4



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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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11 Oct 2011, 05:39
shrive555 wrote: If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy? 1 4 5 7 8 The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance 4^a will end in either 4 or 6 . 9^b will always end in either 9 or1 . So the possible unit digit of xy can be either of one below : 4*9 =6 or 4*1 = 4 OR 6*9 = 4 or 6*1 = 6. So , as you can see the unit digit can either be 6 or 4. We have just 4 in the choices.Hence , option (B) 4 is the correct option.
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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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11 Oct 2011, 23:22
shrive555 wrote: If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy? 1 4 5 7 8 The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance Another thing: \(x = 4^a\) and \(y = 9^b\) When you multiply, \(x*y = 4^a * 9^b\) You cannot bring the powers together. Only if the bases are the same, then you can add the powers. e.g. \(x = 4^a\) and \(y = 4^b\) When you multiply, \(x*y = 4^{a+b}\) Or when the powers are the same, then you can multiply the bases together e.g. \(x = 4^a\) and \(y = 9^a\) When you multiply, \(x*y = 36^a\)
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Re: If a and b are positive integers than x=4^a & y=9^b [#permalink]
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13 Oct 2011, 09:51
VeritasPrepKarishma wrote: shrive555 wrote: If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy? 1 4 5 7 8 The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance Another thing: \(x = 4^a\) and \(y = 9^b\) When you multiply, \(x*y = 4^a * 9^b\) You cannot bring the powers together. Only if the bases are the same, then you can add the powers. e.g.\(x = 4^a\) and \(y = 4^b\) When you multiply, \(x*y = 4^{a+b}\) Or when the powers are the same, then you can multiply the bases together e.g. \(x = 4^a\) and \(y = 9^a\) When you multiply, \(x*y = 36^a\) Thanks for pointing out!!
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